Method and apparatus for configuring quantum mechanical state, and communication method and apparatus using the same

ABSTRACT

If a quantum mechanical state including a plurality of two-level systems (X1, X2, . . . , X2p+1) is expressed by a superposition of orthonormal bases in which each two-level system assumes a basic or an excited state, a quantum gate network is used to perform an operation including a combination of a selective rotation operation and an inversion about average operation D in order to configure a desired partly-entangled quantum mechanical state in which the coefficients of the respective bases are all real numbers.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method and apparatus for configuringa, quantum mechanical state required for quantum information processing,quantum communication, or quantum precision measurements, as well as acommunication method and apparatus using such a method and apparatus.

2. Related Background Art

The fields of quantum computations (see R. P. Feynman, “Feynman Lectureson Computation,” Addison-Wesley (1996)) and quantum information theoriesare advancing rapidly. In these fields, superposition, interference, andan entangled state, which are the basic nature of quantum mechanics, areingeniously utilized.

In the field of quantum computations, since the publication of Shor'salgorithm concerning factorization (see P. W. Shor, “Polynomial-TimeAlgorithms for Prime Factorization and Discrete Logarithms on a QuantumComputer,” LANL quantum physics archive quant-ph/9508027. Similarcontents are found in SIAM J. Computing 26 (1997), 1484. In addition,the first document is P. W. Shor, “Algorithms for quantum computation:Discrete logarithms and factoring,” in Proceedings of the 35th AnnualSymposium on Foundations of Computer Science (ed. S. Goldwasser) 124-134(IEEE Computer Society, Los Alamitos, Calif., 1994). A detaileddescription is found in Artur Ekert and Richard Jozsa, “Quantumcomputation and Shor's factoring algorithm,” Rev. Mod. Phys. 68, 733(1996)) and Grover's algorithm concerning the search problem (L. K.Grover, “A fast quantum mechanical algorithm for database search,” LANLquantum physics archive quant-ph/9605043. Almost similar contents arefound in L. K. Grover, “Quantum Mechanics Helps in Searching for aNeedle in a Haystack,” Phys. Rev. Lett. 79, 325 (1997)), manyresearchers have been proposing methods for implementing quantumcomputations and developing new quantum algorithms.

On the other hand, in the field of quantum information theories, theentangled state has been known to play an important role due to itsunlikelihood to be affected by decoherence (see C. H. Bennett, C. A.Fuchs, and J. A. Smolin, “Entanglement-Enhanced Classical Communicationon a Noisy Quantum Channel,” Quantum Communication, Computing, andMeasurement, edited by Hirota et al., Plenum Press, New York, p. 79(1997)).

Furthermore, as an application of these results, a method for overcomingthe quantum shot noise limit using (n) two-level entangled states hasbeen established through experiments on Ramsey spectroscopy (see D. J.Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heizen,“Spin squeezing and reduced quantum noise in spectroscopy,” Phys. Rev. A46}, R6797 (1992) or D. J. Wineland, J. J. Bollinger, W. M. Itano, andD. J. Heizen, “Squeezed atomic states and projection noise inspectroscopy,” Phys. Rev. A 50, 67 (1994)). Despite the lack ofdiscussion of the two-level Ramsey spectroscopy, a similar concept isdescribed in M. Kitagawa and M. Ueda, “Nonlinear-InterferometricGeneration of Number-Phase-Correlated Fermion States,” Phys. Rev Lett.67, 1852 (1991).

If decoherence in the system caused by the environment is negligible,the maximally entangled state serves to improve the accuracy inmeasuring the frequency of an energy spectrum.

In this case, the fluctuation of the frequency decreases by 1/{squareroot over (n)}. With decoherence in the system considered, however, theresolution achieved by the maximally entangled state is only equivalentto that achieved by an uncorrelated system.

In addition, the use of a partly entangled state having a high symmetryhas been proposed in S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K.Ekert, M. B. Plenio, J. I. Cirac, “Improvement of Frequency Standardswith Quantum Entanglement,” Phys. Rev. Lett. 79, 3865 (1997).

If optimal parameters (coefficients of basic vectors) can be selectedbeforehand, this method can provide a higher resolution than themaximally entangled or uncorrelated state.

The partly entangled state having a high symmetry is given by thefollowing equation: $\begin{matrix}{{{\psi_{n}}\rangle} = {{\sum\limits_{k = 0}^{\lfloor{n/2}\rfloor}\quad {a_{k}{{k}\rangle}_{s}\quad {for}\quad n}} \geqq 2}} & (1)\end{matrix}$

where (n) represents the number of qubits that are twolevel particlesconstituting a state, and ┐n/2┘ represents a maximum integer not morethan n/2. {a_(k)} is a real number wherein, for example, an optimalcombination of values are assumed to be provided beforehand so as toprovide a high resolution in the Ramsey spectroscopy. In this case,{a_(k)} may be a constant.

|k>_(s) is a superposition of states in which (k) or (n−k) qubits areexcited, wherein the superposition is established using an equal weight.For example, |ψ₄> is given by the following equation. $\begin{matrix}{{{{{{{{\psi_{4}}\rangle} = \quad {a_{0}{0}}}\rangle} + {a_{1}{1}}}\rangle} + {a_{2}{2}}}\rangle} \\{= \quad {\left. {{{{a_{0}\left( {0000} \right.}\rangle} + {1111}}\rangle} \right) +}} \\{\quad {a_{1}\left( {{{{{{{{{{{{{0001}\rangle} + {0010}}\rangle} + {0100}}\rangle} + {1000}}\rangle} + {1110}}\rangle} + {1101}}\rangle} +} \right.}} \\{{\quad \left. {{{{1011}\rangle} + {0111}}\rangle} \right)} +} \\{\quad \left. {{{{{{{{{{{{a_{2}\left( {0011} \right.}\rangle} + {0101}}\rangle} + {0110}}\rangle} + {1001}}\rangle} + {1010}}\rangle} + {1100}}\rangle} \right)}\end{matrix}$

These states have symmetry such as that described below.

Invariable despite the substitution of any two qubits

Invariable despite the simultaneous inversion of {|0>, |1>} for eachqubit.

To conduct experiments on the Ramsey spectroscopy using a partlyentangled state having a high symmetry, a target entangled state must beprovided as soon as possible as an initial system state beforedecoherence may occur. Thus, all actual physical systems have adecoherence time, and quantum mechanical operations must be performedwithin this time. This is a problem in not only quantum precisionmeasurements but also quantum communication and general quantumcomputations.

In addition, to configure the target entangled state from a specifiedstate, an operation using basic quantum gates must be performed out manytimes. Minimizing this number of times leads to the reduction of thetime required to provide the target entangled state.

In addition, to configure a quantum gate network for obtaining thetarget entangled state, a conventional computer must be used todetermine in advance which qubits will be controlled by the basicquantum gates, the order in which the basic quantum gates will be used,and a rotation parameter for unitary rotations. The amount of thesecomputations is desirably reduced down to an actually feasible level.

SUMMARY OF THE INVENTION

Thus, an object of the present invention is to provide a method used toconfigure a target entangled state within a decoherence time in anactual physical system, for configuring the desired state using as lesssteps as possible if the computation time is evaluated based on thetotal number of basic quantum gates.

Another object of the present invention is to provide an effectiveprocedure used in configuring a network consisting of basic quantumgates, for using a conventional computer to determine in advance whichqubits will be controlled by the basic quantum gates, the order in whichthe basic quantum gates will be used, and a unitary rotation parameter.

Yet another object of the present invention is to provide a method andapparatus for configuring not only a partly entangled state having ahigh symmetry but also a partly entangled state defined by a functionwith an even number of collisions, using as less steps as possible ifthe computation time is evaluated based on the total number of basicquantum gates.

According to one aspect, the present invention which achieves theseobjects relates to a method for configuring a quantum mechanical stateconsisting of a plurality of two-level systems wherein if asuperposition of orthonormal bases in which each two-level systemassumes a basic or an excited state is used for expression, a desiredpartly-entangled quantum mechanical state in which the coefficients ofthe bases are all real numbers is configured using an operation that isa combination of a selective rotation operation and an inversion aboutaverage operation.

According to one aspect, the present invention which achieves theseobjectives relates to a state configuration apparatus comprising aselective rotation operation means for a plurality of two-level systemsand an inversion about average operation means for the plurality oftwo-level systems, wherein if a quantum mechanical state consisting ofthe plurality of two-level systems is expressed by a superposition oforthonormal bases in which each two-level system assumes a basic or anexcited state, a desired partly-entangled quantum mechanical state inwhich the coefficients of the bases are all real numbers is configuredusing an operation that is a combination of an operation performed bythe selective rotation operation means and an operation performed by theinversion about average operation means.

Other objectives and advantages besides those discussed above shall beapparent to those skilled in the art from the description of a preferredembodiment of the invention which follows. In the description, referenceis made to accompanying drawings, which form a part thereof, and whichillustrate an example of the invention. Such an example, however, is notexhaustive of the various embodiments of the invention, and thereforereference is made to the claims which follow the description fordetermining the scope of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a Feynman diagram of a quantum gate network showing a case inwhich the “first trial method” does not use a second register for n=2;

FIG. 2 is a Feynman diagram of the quantum gate network showing a casein which the “first trial method” does not use the second register forn=3;

FIG. 3 is a Feynman diagram of the quantum gate network showing a casein which the “first trial method” does not use the second register forn=4;

FIG. 4 is a Feynman diagram of the quantum gate network showing a casein which the “first trial method” does not use the second register forn=4;

FIG. 5 is a Feynman diagram of the quantum gate network showing a casein which the “first trial method” does not use the second register forn=4;

FIG. 6 shows the overall flow of the “first trial method” processing fora general (n);

FIG. 7 is a Feynman diagram of quantum gates that selectively rotate aparticular state of a first register;

FIGS. 8A, 8B and 8C show the variation of the coefficients of basicvectors when a quantum mechanical state is subjected to an inversionabout average operation D and an operation for selectively rotating thephase by π;

FIG. 9 shows a mapping caused by a function (f) having an even number ofcollisions;

FIG. 10 is a Feynman diagram of a quantum gate network for the function(f);

FIG. 11 describes an operation of an auxiliary qubit;

FIG. 12 is a Feynman diagram of a program QUBIT-ADDER2.

FIG. 13 is a Feynman diagram showing how QUBIT-ADDER2 is repeatedlyoperated to configure QUBIT-ADDER1;

FIG. 14 is a Feynman diagram showing a quantum gate network with aselective rotation in the second registers;

FIG. 15 is a Feynman diagram showing a network of quantum gates thatexecutes a selective rotation in the second register for η=η−η=n/2;

FIG. 16 is a Feynman diagram showing a network for D;

FIG. 17 is a Feynman diagram showing a network for a subroutine (R(π)D).

FIG. 18 is a Feynman diagram showing a selective π phase rotation in thefirst: register;

FIG. 19 is a Feynman diagram of the quantum gates in a case in which|ψn> can be configured using merely a (R′{η}DR{η}) contraction methodwithout a (R(π)D) repetition operation;

FIG. 20 is a Feynman diagram showing a method for dividingΛ_(n)(R_(Z)(α)) into simpler quantum gates;

FIG. 21 is a Feynman diagram showing a conventional processing for n=2;

FIG. 22 is a Feynman diagram showing a method operative on a 7-quibitnetwork for dividing Λ₅(σ_(x)) into simpler quantum gates;

FIG. 23 is a Feynman diagram showing a method operative on a(n+1)-qubit: network for dividing Λ_(n−1)(σ_(x)) into simpler quantumgates for n≧7;

FIG. 24 is a Feynman diagram showing a method operative on the(n+1)-qubit network for dividing Λ_(m)(σ_(x)) into simpler quantum gatesfor n≧7;

FIG. 25 is a table showing into how many basic gates Λ_(n−1) (σ_(x)) canbe divided on the (n+1)-qubit network;

FIG. 26 is a table showing into how many basic gates Λ_(n)(R_(Z)(α)) canbe divided on the (n+1)-qubit network; and

FIG. 27 shows an example of a quantum communication apparatus using amethod and apparatus for configuring an entangled state according to thepresent invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

One preferred embodiment of the present invention will be described withreference to the accompanying drawings.

(Basic Ideas Common to Each Embodiment)

This embodiment is described in the following order.

1. Simple method (“first trial method”)

2. Specific example of the “first trial” method

3. Method for configuring a gate network for the first trial method ifsecond registers are not used.

4. Conditions under which the (R′{η}DR{η}) contraction method issuccessful

5. (R(π)D) repetition method

6. Simple example of the R(π)D) repetition method

7. Configuration of a partly entangled state using a function with aneven number of collisions

8. Example of configuration of a specific quantum gate network and theevaluation of computation time

(First embodiment) Quantum gate network for a function with an evennumber of collisions

(Second embodiment) Quantum gate network for selective phase rotations

(Third embodiment) Quantum gate network for inversion about averageoperations

(Fourth embodiment) Evaluation of the amount of computations during theoverall process

(Fifth embodiment) Method for configuring a Λ_(n)(R_(Z)(α) gate

(Sixth embodiment) Method implemented using the cold trapped ion method

(Seventh embodiment) Method applied to quantum communication

These embodiments are sequentially described below.

[Simple Method (First Trial Method)]

First, the following method is considered as one for configuring apartly entangled state having a high symmetry. For convenience, thismethod is referred to as the “first trial method” herein. Although shownbelow, this method can configure |ψ_(n)> specified by a set of arbitraryreal coefficients {a_(k)} for n=2, 3 but only in special cases for n≧4.

The “first trial method,” however, is a basic concept for all theembodiments, and the embodiments have been proposed as improved versionsof the first trial method.

(First Trial Method)

1. The following initial state is prepared in an n-qubit register.|s₁>=|0 . . . 0>.

2. Hadamard transformation is applied to each qubit of the register. TheHadamard transformation is given by the following equation.$\begin{matrix}{H = {\frac{1}{\sqrt{2}}\left\lbrack {\begin{matrix}1 \\1\end{matrix}\begin{matrix}1 \\{- 1}\end{matrix}} \right\rbrack}} & (2)\end{matrix}$

In the expression in Equation (2), |0>=(1, 0), |1>=(0, 1). Thisoperation provides a flat superposition in the register as shown by thefollowing equation. $\begin{matrix}{{{{s_{2}}\rangle} = {\frac{1}{\sqrt{2^{n}}}{\sum\limits_{k = {\{{0,1}\}}^{n}}\quad {k}}}}\rangle} & (3)\end{matrix}$

3. s(k) is defined to be the total number of qubits excited at|k. Whenp=└n/2┘, a permutation {η₀, η₁, . . . η_(p−1)} is assumed to be obtainedby selecting (p) elements from (0, 1, . . .p and placing them in anappropriate order (thus, η_(i) is an integer that meets 0≦η_(i)≦p for0≦i≦p, and η _(i)≠η_(j) for i≠j). The following three operations (a),(b), and (c) are sequentially performed for i=0, 1, . . . p−1.

(a) An appropriate 0≦φ_(i)<2π is selected. If s(k) is equal to η_(i), aselective rotation is executed to rotate the phase of |k> by φ_(i). Ifs(k) is equal to (n−η_(i)), the phase of |k> is rotated by (−φ_(i)).

(b) Grover's inversion about average operation D is performed on theregister (see L. K. Grover, “A fast quantum mechanical algorithm fordatabase search,” LANL quantum physics archive quant-ph/9605043 and L.K. Grover, “Quantum Mechanics Helps in Searching for a Needle in aHaystack,” Phys. Rev. Lett. 79, 325 (1997))

(c) If s(k) is equal to η_(i), the phase of |k> is rotated by θ_(i). Ifs(k) is equal to (n−η_(i)), a selective rotation is executed to rotatethe phase of |k> by (−θ_(i)). A phase rotation parameter θ_(i) isselected so as to offset the phase of a basic vector to make thecoefficient a real number.

In the above description, the permutation (η_(o), η₁, . . . η_(p−i)} andphase rotation parameter {θ₀, θ₁, . . . θ_(p−1), depend on the set ofcoefficients {a_(k)} in the Equation (1).

If (n) is an even number and η=n−η=n/2, the number of basic vectors withwhich s(k)=η is established is even. Thus, in the above step 3(a), thephase is rotated by θ for half of all the basic vectors with whichs(k)=η is established, whereas the phase is rotated by (−θ) for theremaining half of the basic vectors. Similar operations are performed instep 3(c).

To execute the selective rotation efficiently, two registers areprovided. A first register consists of (n) qubits and a second registerconsists of ┌log₂(n+1)┐ qubits. ┌log₂(n+1)┐ represents a minimum integerlarger than or equal to log₂(n+1). The Hadamard transformation isapplied to each qubit of the first register to obtain the flatsuperposition, and then the value of s(k) is written to the secondregister as shown in the following expression. $\begin{matrix}{{\left. {{{{\frac{1}{\sqrt{2^{n}}}{\sum\limits_{k = {\{{0,1}\}}^{n}}\quad {k}}}\rangle} \otimes {0}}\rangle}\rightarrow{\frac{1}{\sqrt{2^{n}}}{\sum\limits_{k = {\{{0,1}\}}^{n}}{k}}} \right.\rangle} \otimes {{{{s(k)}}\rangle}.}} & (4)\end{matrix}$

When the selective rotation is executed, the number of quantum gatesused can be saved by setting in the second register a control sectionfor the quantum gates. For example, consider a case in which n=4 ands(k)=1. The rotation of the phase of the second register in the state of|1> is equal to the rotation of the phases of eight basic vectors in thefirst vector. The second register must be initialized to |0 . . . 0>before the inversion about average operation D is performed and before afinal state is obtained.

To confirm the operation of the method described above, the status |s₃>after step 3(a) (i=0) is expressed as follows. $\begin{matrix}\begin{matrix}{{{{s_{3}}\rangle}\quad = {R\left\{ \eta \right\} {s_{2}}}}\rangle} \\{\quad {= {{\frac{1}{\sqrt{2^{n}}}\underset{h}{\underset{}{\left\lbrack {{\exp \left( {i\quad \varphi} \right)},\ldots \quad,} \right.}}\underset{h}{\underset{}{{\exp \left( {{- i}\quad \varphi} \right)},\ldots \quad,}}1},\ldots \quad,{1\underset{\underset{2^{n} - {2h}}{}}{\rbrack}}}}}\end{matrix} & (5)\end{matrix}$

where (h) is the number of basic vectors for which s(k) is equal to η ish.

R{η}) represents the selective rotation used in step 3(a). In this case,i=0 that is a subscript has been omitted. In Equation (5),{|k>|kε{0,1}^(n)} is used as an orthonormal basis. As in the expressionin Equation (5), the embodiments often describe vertical vectors ashorizontal vectors. In addition, the embodiments represent (h) as thenumber of basic vectors for which s(k) is Equal to η. Thus, (h)components exp(iφ) and (h) components exp(−iφ) exist in |s₃>.

In the expression in (5), the order of the basic vectors {|0 . . . 00>,|0 . . . 01> . . . , |1 . . . 11>} has been changed so that thecomponents exp (iφ) and exp(−iφ) are collected at the left end of |s₃>.Such a change in the order of the basic vectors does not change thematrix expression of the inversion about average operation D. A2^(n)×2^(n) matrix expression of D is given as shown below.$\begin{matrix}\left\{ \begin{matrix}{D_{ij} = 2^{{- n} + 1}} & {{{if}\quad i} \neq j} \\{D_{ii} = {{- 1} + 2^{{- n} + 1}}} & \quad\end{matrix} \right. & (6)\end{matrix}$

Applying D to |s₃> results in the following state:

|s ₄ >=D|s ₃>=[α, . . . , α′, . . . β, . . . ],  (7)

where α and β meet the following conditions. $\begin{matrix}\left\{ \begin{matrix}{{{{2^{{({3{n/2}})} - 1}\alpha} = {{2h\quad \cos \quad \varphi} + \left( {2^{n} - {2h}} \right) - {2^{n - \quad 1}e^{i\quad \varphi}}}},}\quad} \\{{2^{{({3{n/2}})} - 1}\beta} = {{2h\quad \cos \quad \varphi} + \left( {2^{n} - {2h}} \right) - 2^{n - \quad 1}}}\end{matrix} \right. & (8)\end{matrix}$

To offset the phase, θ is defined as follows: $\begin{matrix}{e^{i\quad \theta} = \frac{\alpha\bullet}{\alpha }} & (9)\end{matrix}$

Next, selective rotation R′ {η} acts upon |s₄>. $\begin{matrix}{{{{{{s_{5}\rangle} = {R^{\prime}\left\{ \eta \right\}}}}s_{4}}\rangle} = {\left\lbrack {{e^{i\quad \theta}\alpha},\ldots \quad,{e^{{- i}\quad \theta}\alpha^{\prime}},\ldots \quad,\beta,\ldots}\quad \right\rbrack \quad = {\left\lbrack {{\underset{2h}{\underset{}{{\alpha },\ldots \quad,}}\beta},\ldots}\quad \right\rbrack.}}} & (10)\end{matrix}$

where the number of components |α| is 2h. To make the coefficients of 2hbasic vectors negative, the phases of the 2h basic vectors may berotated selectively by π.

For example, for n=2, |ψ₂> is given as follows:

|ψ>₂ =a ₀(|00>+|11>)+a ₁(|01>+|10>),  (11)

where the following relations are established. $\begin{matrix}\left\{ \begin{matrix}(i) & {{{1/2} \leqq a_{0} \leqq {1/\sqrt{2}}},} & {{0 \leqq {a_{1}} \leqq {1/2}},} \\({ii}) & {{0 \leqq {a_{0}} \leqq {1/2}},} & {{1/2} \leqq a_{1} \leqq {1/{\sqrt{2}.}}}\end{matrix} \right. & (12)\end{matrix}$

By executing the above method using the operation for rotating the phaseby (±φ) for |00> and |11> in which 2-qubits are flatly superposed on oneanother, |ψ₂> is obtained as shown by the following equation.$\begin{matrix}{{a_{0} = \sqrt{\frac{1 + {\sin^{2}\varphi}}{2}}},\quad {a_{1} = \frac{\cos \quad \varphi}{2}}} & (13)\end{matrix}$

This is included within the range of 12(i). φ must be determined from(13) using conventional computations before quantum operations areperformed.

The “first trial method” enables the configuration of |ψ₂> and |ψ₃> thatare provided by arbitrary real coefficients. For n≧4, however, |ψ_(n)>that cannot be configured by this method exists depending on the set ofcoefficients {a_(k)}. For example, the following state cannot beconfigured by this method.${{\psi_{4}}\rangle} = {{\sqrt{\frac{3}{7}}{0\rangle}_{s}} + {\frac{1}{7\sqrt{2}}{1\rangle}_{s}} + {\frac{1}{7\sqrt{2}}{{2\rangle}_{s}.}}}$

The “first trial method” corresponds to continuous execution of the(R′{η}DR{η}) contraction method, which is described below. The(R′{η}DR{η}) contraction method is not always successful, but there aresufficient conditions for success, which will also be described below.

In addition, as described for Equation (13), the “first trial method”must use conventional computations to determine the phase rotationparameter φ_(n) before unitary gates are operated. The needs forconventional computations prior to the configuration of a quantum gatenetwork are common to all embodiments.

According to the “first trial method,” (p+1) basic vectors are availablefor selective rotations for p=└n/2┘. Thus, there are (p+1)! permutationsto which (R′{η}DR{η}) is applied. If, however, |ψ_(r)> can beconfigured, (R′{η}DR{η}) is applied to only some of these permutations.Furthermore, certain |ψ_(n)> cannot be configured by this method.

In addition, trials and errors must be carried out (p+1)! times usingconventional computations before the phase rotation parameter can bedetermined. Thus, the burden of computations becomes heavier as (n)increases.

Thus, as a method for solving these problems, a method will be describedbelow that uses sufficient conditions for the success of the(R′{η}DR{η}) contraction method and a (R(π)D) repetition methodaccording to the embodiments.

[Specific Example of the “First Trial Method”]

Before explaining the sufficient conditions for the success of the(R′{η}DR{η}) contraction method and the (R(π)D) repetition method,descriptions are given of changes in qubit for n=2, 3, 4 according tothe “first trial method” and of a quantum gate network configuration ina case where the second register is not used.

(n) and (a₀, a₁, . . . , a_(p)} are assumed to be given beforehand,where p=└n/2┘. A quantum gate network is considered that configures thepartly entangled state with a high symmetry |ψ_(n)> specified by theseconstants and expressed by Equation (1).

To examine specific changes in qubit state, a case of n=2 is consideredas a first simple example. In the meantime, the following relations areassumed in Equation (11).

½≦a ₀≦1/{square root over (2)}, −½≦a ₁≦½

First, as an initial state, |00> is provided for a first and a secondqubits, and H (Hadamard transformation) is individually applied to eachqubit. This operation transforms the qubits to the following state.

|00>→½(|00>+|11>+|10>+|11>)

A selective rotation is executed in such a way that |00> and |11> areeach multiplied by phase factor exp(iφ). The resulting state of thefirst and second qubits is denoted as |ψ>.

|ψ>=½(e ^(iφ)|00>+|01>+|10>+e ^(−iφ)|11>)

Next, an inversion about average operation is performed. Since n=2, D isgiven in the form of a matrix of 4×4, using Equation (6).$D = {\frac{1}{2}{\begin{matrix}{- 1} & 1 & 1 & 1 \\1 & {- 1} & 1 & 1 \\1 & 1 & {- 1} & 1 \\1 & 1 & 1 & {- 1}\end{matrix}}\begin{matrix}{{00}\rangle} \\{{01}\rangle} \\{{10}\rangle} \\{{11}\rangle}\end{matrix}}$   ⟨00    ⟨01  ⟨10    ⟨11

The application of D to |ψ> leads to the following transformation.${\psi\rangle} = {\left. {\frac{1}{2}\begin{bmatrix}{\exp \left( {i\quad \varphi} \right)} \\1 \\1 \\{\exp \left( {{- i}\quad \varphi} \right)}\end{bmatrix}}\rightarrow{D{\psi\rangle}} \right. = {\frac{1}{2}\begin{bmatrix}{1 - {i\quad \sin \quad \varphi}} \\{\cos \quad \varphi} \\{\cos \quad \varphi} \\{1 + {i\quad \sin \quad \varphi}}\end{bmatrix}}}$

Next, |00> and |11> are again multiplied by phase factors exp(iθ) andexp(−iθ), respectively. In this case, θ meets the following relation.$e^{i\quad \theta} = \frac{1 + {i\quad \sin \quad \varphi}}{{1 + {i\quad \sin \quad \varphi}}}$

Such a selective rotation causes the two qubits to enter the followingstate.${\frac{\sqrt{1 + {\sin^{2}\varphi}}}{2}\left( {{00\rangle} + {{11}\rangle}} \right)} + {\frac{\cos \quad \varphi}{2}\left( {{01\rangle} + {{10}\rangle}} \right)}$

Thus, by selecting beforehand φ to meet the following equation,${a_{0} = \frac{\sqrt{1 + {\sin^{2}\varphi}}}{2}},\left( {a_{1} = \frac{\cos \quad \varphi}{2}} \right)$

the target partly entangled state |ψ₂> is obtained.

The following point is noted. Since the physical quantity of |ψ₂> doesnot vary despite a rotation of the phase of the entire |ψ₂>, the rangesof the values of a₀ and a₁ are as shown below.

½≦a ₀≦1/{square root over (2)}, 0≦|a ₁|≦½  (i)

0≦|a ₀|≦½, ½≦a ₁≦1/{square root over (2)}  (ii)

The above specific example provides a method for configuring |ψ₂> thatmeets the condition in (i). To configure |ψ₂> that meets the conditionin (ii), the selective rotation may be applied to |01>and |10>instead of|00>and |11>. Thus, for n=2, an arbitrary |ψ₂> can be configured usingthe “first trial method.”

Next, a case of n=3 is considered.

|ψ₃ >=a ₀(|000>+|111>)+a₁(|001>+|010>+|100>+|110>+|101>+|011>)  (14)

In this equation, a₀ ²+3a₁ ²=½. In the meantime, the following relationsare assumed.

½{square root over (2)}≦a ₀≦1/{square root over (2)}, 0≦a ₁≦½{squareroot over (2)}

First, as an initial state, |000> is provided for a first, a second, anda third qubits, and H (Hadamard transformation) is applied to each qubitindependently.$\left. {000\rangle}\rightarrow{\frac{1}{2\sqrt{2}}\left( {{000\rangle} + {001\rangle} + {010\rangle} + {011\rangle} + {100\rangle} + {\quad 101\rangle} + \quad {110\quad\rangle} + \quad {111\quad\rangle}} \right)} \right.$

A selective rotation is executed in such a way that |000> and |111> aremultiplied by phase factor exp(iφ) and exp(−iφ), respectively. Then, thefollowing state |ψ> is obtained.${\psi\rangle} = {\frac{1}{2\sqrt{2}}\left( {{^{\quad \varphi}{000\rangle}} + {001\rangle} + {010\rangle} + {011\rangle} + {100\rangle} + {\quad 101\rangle} + \quad {110\quad\rangle} + \quad {^{{- }\quad \varphi}\quad {111\rangle}}} \right)}$

Next, the inversion about average operation D is performed. If n=3 inEquation (6), the following matrix expression is given. $\begin{matrix}{D = {\frac{1}{4}{\begin{matrix}{- 3} & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\1 & {- 3} & 1 & 1 & 1 & 1 & 1 & 1 \\1 & 1 & {- 3} & 1 & 1 & 1 & 1 & 1 \\1 & 1 & 1 & {- 3} & 1 & 1 & 1 & 1 \\1 & 1 & 1 & 1 & {- 3} & 1 & 1 & 1 \\1 & 1 & 1 & 1 & 1 & {- 3} & 1 & 1 \\1 & 1 & 1 & 1 & 1 & 1 & {- 3} & 1 \\1 & 1 & 1 & 1 & 1 & 1 & 1 & {- 3}\end{matrix}}\begin{matrix}{\quad {000\rangle}} \\{\quad {001\rangle}} \\{\quad {010\rangle}} \\{\quad {011\rangle}} \\{\quad {100\rangle}} \\{\quad {101\rangle}} \\{\quad {110\rangle}} \\{\quad {111\rangle}}\end{matrix}}} \\\begin{matrix}{\quad {{\langle 000}\quad {\langle 001}\quad {\langle 010}\quad {\langle 011}\quad {\langle 100}\quad {\langle 101}\quad {\langle 110}\quad {\langle 111}}} & \quad & \quad & \quad & \quad & \quad & \quad & \quad\end{matrix}\end{matrix}$

The application of D to |ψ> leads to the following transformation.${D{\psi\rangle}} = {\frac{1}{8\sqrt{2}}\begin{bmatrix}{6 - {3^{\quad \varphi}} + ^{{- }\quad \varphi}} \\{2\left( {1 + {\cos \quad \varphi}} \right)} \\{2\left( {1 + {\cos \quad \varphi}} \right)} \\{2\left( {1 + {\cos \quad \varphi}} \right)} \\{2\left( {1 + {\cos \quad \varphi}} \right)} \\{2\left( {1 + {\cos \quad \varphi}} \right)} \\{2\left( {1 + {\cos \quad \varphi}} \right)} \\{6 - {3^{{- }\quad \varphi}} + ^{\quad \varphi}}\end{bmatrix}}$

Next, |000> and |111> are again multiplied by phase factors exp(iθ) andexp(−iθ). In this case, θ meets the following relation.$^{\quad \varphi} = \frac{6 - {3^{{- }\quad \varphi}} + ^{\quad \varphi}}{{6 - {3^{\quad \varphi}} + ^{{- }\quad \varphi}}}$

Such a selective rotation causes the two qubits to enter the followingstate. $\begin{matrix}{{\frac{\sqrt{23 - {12\cos \quad \varphi} - {3\cos \quad 2\varphi}}}{8}\left( {{00\rangle} + {11\rangle}} \right)} + {\frac{1 + {\cos \quad \varphi}}{4\sqrt{2}}\left( {{01\rangle} + {10\rangle}} \right)}} & (15)\end{matrix}$

Thus, by selecting beforehand φ to meet the following equation,${a_{0} = \frac{\sqrt{23 - {12\cos \quad \varphi} - {3\cos \quad 2\varphi}}}{8}},\left( {a_{1} = \frac{1 + {\cos \quad \varphi}}{4\sqrt{2}}} \right)$

the target partly entangled state |ψ₃> is obtained.

The following point is noted. Based on Equation (15), a₀ and a₁ inEquation (14) each assume the following four values.

½{square root over (2)}≦a ₀≦1/{square root over (2)}, 0≦a ₁≦½{squareroot over (2)}  (i)

½{square root over (2)}≦a ₀≦1/{square root over (2)}, −½{square rootover (2)}≦a ₁≦0  (ii)

0≦a ₀≦½{square root over (2)}, ½{square root over (2)}≦a ₁≦1/{squareroot over (6)}  (iii)

½{square root over (2)}≦a ₀≦0, ½{square root over (2)}≦a ₁≦1/{squareroot over (6)}  (iv)

The above specific example provides a method for configuring |ψ₃> thatmeets the condition in (i). To configure ψ₃> that meets the condition in(ii), |000> and |111> may be multiplied by −exp(iθ) and −exp(−θ) in thesecond selective rotation, respectively. In addition, by subjecting|001>, |010>, |100>, as well as |110>, |101>, and |011> to a selectiverotation,$\left. {{\frac{1}{2\sqrt{2}}\left\lbrack {\quad {000\rangle}} \right.} + {^{\quad \varphi}\left\{ \quad {{001\rangle} + \quad {010\rangle} + \quad {100\rangle}} \right\}} + {^{{- }\quad \varphi}\left\{ \quad {{110\rangle} + \quad {101\rangle} + \quad {011\rangle}} \right\}} + \quad {111\rangle}} \right\rbrack$

then the inversion about average operation D, and$\frac{1}{8\sqrt{2}}\left\lbrack \quad {{2\left( {{3\cos \quad \varphi} - 1} \right)\left( \quad {{000\rangle} + \quad {111\rangle}} \right)} + {\left( {2 - ^{\quad \varphi} + {3^{{- }\quad \varphi}}} \right)\left( \quad {{001\rangle} + \quad {010\rangle} + \quad {100\rangle}} \right)} + {\left( {2 - ^{{- }\quad \varphi} + {3^{\quad \varphi}}} \right)\left( \quad {{110\rangle} + \quad {101\rangle} + \quad {011\rangle}} \right)}} \right\rbrack$

then a selective rotation again, the following relation is obtained.${\frac{{3\quad \cos \quad \varphi} - 1}{4\sqrt{2}}\left( \quad {{000\rangle} + \quad {111\rangle}} \right)} + {\frac{\sqrt{7 + {4\quad \cos \quad \varphi} - {3\quad \cos \quad 2\quad \varphi}}}{8}\left( \quad {{001\rangle} + \quad {010\rangle} + \quad {100\rangle} + \quad {110\rangle} + \quad {101\rangle} + \quad {011\rangle}} \right)}$

This result indicates that the elementary computations have been used toconfigure |ψ₃> that meets the condition in (iii) within the range of0≦φ≦ζ for cos ζ=⅓. |ψ₃> that meets the condition in (iv) can besimilarly configured. Consequently, the “first trial method” canconfigure an arbitrary |ψ₃>.

Next, a case of n=4 is considered.

|ψ₄ >=a ₀(|0000>+|1111>)+a₁(|0001>+|0010>+|0100>+|1000>+|1110>+|1101>+|1011>+|0111>)+a₂(|0011>+|0101>+|0110>+|1001>+|1010>+|1100>)

In this equation, a₀ ²+4a₁ ²+3a₂ ²=½.

First, as an initial state, |0000> is provided for a first, a second, athird, and a fourth qubits, and H (Hadamard transformation) is appliedto each qubit independently.$\left. {{0000}\rangle}\rightarrow{\frac{1}{4}{\sum\limits_{i = 0}^{15}{\underset{\_}{i}\rangle}}} \right.$

In the above expression, i is a binary expression of i. By selectivelymultiplying |0000> and |1111> by phase factor exp(iφ) and exp(−iφ),respectively, the following state |ψ> is obtained.

|ψ>=¼(e ^(iφ)|0>+|1>+ . . . |14>+e ^(−iφ)|15>)

Next, the inversion about average operation D is performed. If n=4,Equation (6) can be written as follows using a matrix expression.$D = {\frac{1}{4}\begin{bmatrix}{- 7} & 1 & 1 & \cdots & 1 \\1 & {- 7} & 1 & \cdots & 1 \\\quad & \cdots & \cdots & \quad & \quad \\\quad & \cdots & \cdots & \quad & \quad \\1 & \cdots & \cdots & \quad & {- 7}\end{bmatrix}}$

The application of D causes the qubits to enter the following state.${{D{\psi}}\rangle} = {\frac{1}{32}\begin{bmatrix}{14 - {7^{\quad \varphi}} + ^{{- }\quad \varphi}} \\{2\left( {3 + {\cos \quad \varphi}} \right)} \\{2\left( {3 + {\cos \quad \varphi}} \right)} \\\cdots \\{14 - {7^{{- }\quad \varphi}} + ^{\quad \varphi}}\end{bmatrix}}$

Next, |0000> and |1111> are again multiplied by phase factors exp(iθ)and exp(−iθ). In this case, e meets the following equation.$^{\quad \varphi} = \frac{14 - {7^{{- }\quad \varphi}} + ^{\quad \varphi}}{{14 - {7^{\quad \varphi}} + ^{{- }\quad \varphi}}}$

Such a selective rotation causes the two qubits to enter the followingstate.${\frac{r_{1}(\varphi)}{32}\left. {{{\left( {0000} \right.\rangle} + {1111}}\rangle} \right)} + {\frac{r_{2}(\varphi)}{32}\left. {{{\left( {0001} \right.\rangle} + \ldots + {1110}}\rangle} \right)}$

In this case, the following equations are established.

r ₁(φ)=|14−7e ^(iφ) +e ^(−iφ) |, r ₂(φ)=2(3+cos φ)

Furthermore, |0011>, |0101>, and |0110 are multiplied by phase factorexp(iδ) and |1001>, |1010>, and |1100> are multiplied by phase factorexp(−iδ). Subsequently, another application of D causes the qubits toenter the following state. $\left. {\frac{1}{32}\begin{bmatrix}{r_{1}(\varphi)} \\{r_{2}(\varphi)} \\{r_{2}(\varphi)} \\{{r_{2}(\varphi)}\exp \quad \left( {\quad \delta} \right)} \\{r_{2}(\varphi)} \\{{r_{2}(\varphi)}\exp \quad \left( {\quad \delta} \right)} \\{{r_{2}(\varphi)}\exp \quad \left( {\quad \delta} \right)} \\{r_{2}(\varphi)} \\{r_{2}(\varphi)} \\{{r_{2}(\varphi)}\exp \quad \left( {{- }\quad \delta} \right)} \\{{r_{2}(\varphi)}\exp \quad \left( {{- }\quad \delta} \right)} \\{r_{2}(\varphi)} \\{{r_{2}(\varphi)}\exp \quad \left( {{- }\quad \delta} \right)} \\{r_{2}(\varphi)} \\{r_{2}(\varphi)} \\{r_{1}(\varphi)}\end{bmatrix}}\rightarrow{{\frac{1}{256}\begin{bmatrix}{{{- 6}{r_{1}(\varphi)}} + {2{r_{2}(\varphi)}\left( {4 + {3\quad \cos \quad \delta}} \right)}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {8 - {5\quad {\exp \left( {\quad \delta} \right)}} + {3\quad {\exp \left( {{- }\quad \delta} \right)}}} \right)}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {8 - {5\quad {\exp \left( {\quad \delta} \right)}} + {3\quad {\exp \left( {{- }\quad \delta} \right)}}} \right)}} \\{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {8 - {5\quad {\exp \left( {\quad \delta} \right)}} + {3\quad {\exp \left( {{- }\quad \delta} \right)}}} \right)}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {8 - {5\quad {\exp \left( {{- }\quad \delta} \right)}} + {3\quad {\exp \left( {\quad \delta} \right)}}} \right)}} \\{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {8 - {5\quad {\exp \left( {{- }\quad \delta} \right)}} + {3\quad {\exp \left( {\quad \delta} \right)}}} \right)}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {8 - {5\quad {\exp \left( {{- }\quad \delta} \right)}} + {3\quad {\exp \left( {\quad \delta} \right)}}} \right)}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{2{r_{1}(\varphi)}} + {6{r_{2}(\varphi)}\cos \quad \delta}} \\{{{- 6}{r_{1}(\varphi)}} + {2{r_{2}(\varphi)}\left( {4 + {3\quad \cos \quad \delta}} \right)}}\end{bmatrix}}\quad \begin{matrix}{{0000}\rangle} \\{{0001}\rangle} \\{{0010}\rangle} \\{{0011}\rangle} \\{{0100}\rangle} \\{{0101}\rangle} \\{{0110}\rangle} \\{{0111}\rangle} \\{{1000}\rangle} \\{{1001}\rangle} \\{{1010}\rangle} \\{{1011}\rangle} \\{{1100}\rangle} \\{{1101}\rangle} \\{{1110}\rangle} \\{{1111}\rangle}\end{matrix}} \right.$

|0011>, |0101>, and |0110> are multiplied by phase factor exp(iκ) and|1001>, |1010>, and |1100> are multiplied by phase factor exp(−iκ). Inthis case, κ meets the following relation.$^{\quad \kappa} = \frac{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {2 - {5{\exp \left( {{- }\quad \delta} \right)}} + {3{\exp \left( {\quad \delta} \right)}}} \right)}}{{{2{r_{1}(\varphi)}} + {{r_{2}(\varphi)}\left( {8 - {5{\exp \left( {\quad \delta} \right)}} + {3{\exp \left( {{- }\quad \delta} \right)}}} \right)}}}$

Such a selective rotation causes the qubits to enter the followingstate.

{fraction (1/256)}{s ₁(φ,δ)(|0000>+|1111>)+ s ₂(φ,δ)(|0001>+ . . .|0111>)+s ₃(φ,δ)(|0011>+ . . . +|1100>)}

In this case, the following equations are established.

S ₁(φ,δ)=−6r ₁(φ)+2r ₂(φ) (4+3 cos δ)

S ₂(φ,δ)=2r ₁(φ)+6r ₂(φ)cos δ)

S ₃(φ,δ)=|2r ₁(φ)+r ₂(φ)(8−5 exp(iδ)+3 exp(−iδ))|

By selecting appropriate parameters as φ and δ, the particular partlyentangled state |ψ₄> is obtained. Appropriate selection of a term forthe selective rotation enables various combinations of possible valuesof coefficients {a₀, a₁, a₂} to be accommodated. As described above,however, the first trial method” cannot always configure an arbitrary|ψ_(n)> for n≧4.

[Method for Configuring a Gate Network Using the “First Trial Method”without the Second Register]

Next, FIGS. 1, 2, and 3 to 5 show a method for using of the “first trialmethod” to configure a network of quantum gates without the secondregister, for n=2, 3, and 4.

It is determined how the total number of basic quantum gates requiredfor a network of quantum gates varies with an increase in (n) if the“first trial method” does not use the second register. Forsimplification, (n) is assumed to be an odd number, so it can beexpressed as n=2p+1. In this case, Equation (1) is expressed as follows:${\psi_{{2p} + 1}\rangle} = {\sum\limits_{k = 0}^{p}{a_{k}{k\rangle}_{s}}}$

where |k> denotes a superposition of the states of 2_(2p+1)C_(k) termsin the (2p+1) qubits which contain k |1> or (2p+1−k)|1>, wherein thesuperposition is established using an equal weight. In addition theselective rotation is executed in the particular order of |k>(k=1, . . ., p).

FIG. 6 shows the flow of the overall process. X₁, . . . X_(2p+1)represent the (2p+1) qubits of the first register. A series ofoperations including a selective rotation, an inversion about averageoperation, and a selective rotation are continuously performed (q)times. During the selective rotation of φ_(k), the _(2p+1)C_(k) termscontaining k |1> is multiplied by phase factor exp(iφ_(k)), while theremaining _(2p+1)C_(k) terms containing (2p+1−k) |1> is multiplied byphase factor exp(−iφ_(k)).

FIG. 7 shows quantum gates that selectively rotate a single particularstate. X₁, . . . , X_(n) denote the n qubits of the first register. Inaddition, the shaded rectangles in the figure each contain a NOT-gateσ_(x) or an identity transformation I. Whether the rectangle containsthe NOT-gate or the identity transformation depends on k. Thus, clearly,the selective rotation requires one Λ_(n)(R₂(2φ_(k))) at minimum.

An(R₂(24))) is defined in the (second embodiment) of [Example of aconfiguration of a specific quantum gate network and evaluations of thecomputation time] described below. Although shown in the subsequent(fifth embodiment) of [Example of a configuration of a specific quantumgate network and evaluations of the computation time], 8(2n−7) basicgates have been confirmed to be required to configure Λ_(n)(R₂(2φ_(k))).Accordingly, at least 8(2n−7) basic gates are required for the overallselective rotation. 2_(2p+1)C_(k) states are selected for phase factorsexp(iφ_(k)) and exp(−iφ_(k)). Similar operations performed on phasefactors exp(iθ_(k)) and exp(−iθ_(k)). Since the selective rotation iscarried out from k=1 to k=p, at least the following amount ofcomputations is required in total. $\begin{matrix}{{4 \times 8\left( {{2n} - 7} \right){\sum\limits_{k = 1}^{p}{{}_{{2p} + 1}^{}{}_{}^{}}}} = {32\left( {{2n} - 7} \right)\left( {2^{2p} - 1} \right)}} \\{= {32\left( {{2n} - 7} \right)\left( {2^{n - 1} - 1} \right)}}\end{matrix}$

As described above, if the “first trial method” does not use the secondregister, the amount of computations increases in the order of O(2^(n))with increasing n. In fact, a second register can be provided to limitthe total number of basic gates to the order of O(2^(n)/2), as describedbelow.

[Conditions for the Success of the (R′{η}DR{η}) Contraction Method]

The phase rotation parameter may be determined as follows usingconventional computations. The “first trial method” is a combination ofthe selective rotation and the inversion about average operation. Theseare unitary transformations and |ψ_(n)> can be assumed to be inverse toan operation for transformation to the flat superposition shown below.$\frac{1}{\sqrt{2^{n}}}{\sum\limits_{k = {\{{0,1}\}}^{n}}{k\rangle}}$

Fortunately, the inverse transformation of the selective rotation isalso a selective rotation, while the inverse transformation of theinverse about average operation D is also an inverse about averageoperation D.

Thus, an operation for applying transformation (R′DR) to basic vectorshaving factors a_(i) and a_(j) to make new factors after transformationthe same is called a “(R′DR) contraction operation.” The lemma describednext provides a sufficient condition for the (R′DR) contractionoperation. Then,

p=└n/2┘

is assumed and in the meantime, a_(k)≧0 is assumed for k=0, 1, . . . ,p.

Lemma 0.1 [Sufficient Condition for a Contraction Operation]

|ψ> is assumed to be given as follows:${{\Psi\rangle} = \left\lbrack {{\underset{\underset{2h}{}}{{a_{0},\ldots \quad,}\quad}\underset{\underset{2m}{}}{a_{1},\ldots \quad,}\quad a_{2{({h + n})}}},\ldots \quad,a_{N}} \right\rbrack},$

where N=2^(n−1) and 0≦a₀<a₁. The number of components a₀ is assumed tobe 2h while the number of components a, is assumed to be 2m. h≧1, m≧1,and h+m≦2 ^(n−1) are also assumed. The sum of all the coefficients of|ψ> is represented by S.$S = {{2{ha}_{0}} + {2{ma}_{1}} + {\sum\limits_{k = {2{({h + m})}}}^{2^{n} - 1}a_{k}}}$

If the following condition

S−2 ^(n−2)(a ₀ +a ₁)≧0

is met, a (R′DR) contraction method including a selective rotation of 2mbasic vectors can be used to constantly make the same the coefficientsof 2(h+m) basic vectors, which have been a₀ or a₁.

(Proof) When a selective rotation of a rotating angle θ(0≦θ<2π) isapplied to 2m basic vectors having factor a₁, the following state isobtained.${{{R(\theta)}{\Psi\rangle}} = \left\lbrack {{\underset{\underset{2h}{}}{{a_{0},\ldots \quad,}\quad}\underset{\underset{m}{}}{{^{\quad \theta}a_{1}},\ldots \quad,}\quad \underset{\underset{m}{}}{\quad {{^{{- }\quad \theta}\quad a_{1}},\ldots \quad,}}\quad a_{2{({h + m})}}},\ldots \quad,a_{N}} \right\rbrack},$

However, N=2^(n)−1, thus R′(θ)DR(θ))|ψ> can be written as follows.

R′(θ)DR(θ)|ψ>=[A ₀ , . . . , A ₁ , . . . , A _(2(h+m)) , . . . A ₂_(^(n)) ⁻¹],

In this case, the following equations are assumed. $\begin{matrix}\left\{ \begin{matrix}{{2^{n - 1}A_{0}} = {{\left( {{2h} - 2^{n - 1}} \right)a_{0}} + {2{ma}_{1}\cos \quad \theta} + C}} \\{{2^{n - 1}A_{1}} = {{{2{ha}_{0}} + {\left( {m - 2^{n - 1}} \right)a_{1}^{\quad \theta}} + {{ma}_{1}^{{- }\quad \theta}} + C}}} \\{{{2^{n - 1}A_{j}} = {{2{ha}_{0}} + {2{ma}_{1}\cos \quad \theta} - {2^{n - 1}a_{j}} + C}}\quad} \\{{{{for}\quad j} = {2\left( {h + m} \right)}},\ldots \quad,{2^{n} - 1}}\end{matrix} \right. & (16)\end{matrix}$

The following equation is also assumed.$C = {\sum\limits_{k = {2{({h + m})}}}^{2^{n} - 1}a_{k}}$

To make A₀ ² equal to A₁ ², the following function f(θ) is defined.$\begin{matrix}\begin{matrix}{{f(\theta)} = \quad {2^{n - 2}\left( {A_{0}^{2} - A_{1}^{2}} \right)}} \\{= \quad {{\left( {{2{ha}_{0}} + {2{ma}_{1}\cos \quad \theta} + C} \right)\left( {{a_{1}\cos \quad \theta} - a_{0}} \right)} - {2^{n - 2}{\left( {a_{1}^{2} - a_{0}^{2}} \right).}}}}\end{matrix} & (17)\end{matrix}$

If f(θ)=0 is met, A₀ ² is equal to A₁ ². Then, f(0) and f(π/2) areevaluated as shown below. $\begin{matrix}\left\{ \begin{matrix}{{f(0)} = {\left( {a_{1} - a_{0}} \right)\quad\left\lbrack {S - {2^{n - 2}\left( {a_{0} + a_{1}} \right)}} \right\rbrack}} \\{{f\left( {\pi/2} \right)} = {{{- {a_{0}\left( {{2{ha}_{0}} + C} \right)}} - {2^{n - 2}\left( {a_{1}^{2} - a_{0}^{2}} \right)}} < 0}}\end{matrix} \right. & (18)\end{matrix}$

Thus, if S−2^(n−2)(a₀+a₁)≧0 is met, 0≦θ<(π/2) exists that meets A₀ ²=A₁². If A₀ and A₁ have different signs, a selective π rotation may becarried out.

To determine the phase rotation parameter, the following procedure isused. A given state |ψ_(n)> is expressed using (1). In this case,a_(k)≧0 for 0≦k≦p.

a_(min) is the minimum coefficient of {a_(k)}, and a_(min+1) is thesecond smallest coefficient. Thus, 0≦a_(min)<a_(min+1)<a_(j), but a_(j)is an arbitrary coefficient of |ψ_(n)< other than a_(min) and a_(min+1).

The number of different coefficients in {a_(k)} is (P+1), so the timerequired to find a_(min) and a_(min+1) is only O(n) steps. S is the sumof all the coefficients of |ψ_(n)>.

1. If S<2^(n−2) (a_(min)+a_(min+1)) is established, S<2^(n−2)(a_(i)+a_(j)) is established for all (i) and (j). In this case, the(R′DR) contraction method cannot be carried out. Thus, the (R(π)D)repetition method described below must be used.

2. If S≧2^(n−2) (a_(m)in+a_(min+1)) is established, the (R′DR)contraction method can be executed to obtain relation A_(min)²=A_(min+1) ². Since Equation (17) is a quadratic equation of cos θ, itcan provide e efficiently.

In this case, there may be other pairs of coefficients that can becontracted, these coefficients are ignored. After the execution of the(R′DR) contraction method, by selectively rotating by π the phases ofbasic vectors having a negative coefficient, a state in which allcoefficients are positive or zero can be obtained to determine againwhether the condition in lemma 0.1 has been met.

If the (R′DR) contraction method can be executed (p) times for |ψ_(n)>,a flat superposition can be obtained. In contrast, since the quantumgate network is a reversible process, |ψ_(n)> can be configured from theflat superposition using this network. With Equation (16), {A_(i)} canbe calculated within about O(n) steps using a conventional computer.This is because the number of different coefficients in {A_(i)} is(p+1).

[(R(π)D) Repetition Method]

Methods will be considered that can contract coefficients if thecondition in lemma 0.1, that is, S≧2^(n−2) (a_(min)+a_(min+1)) is notmet. For example, the following state is assumed. $\begin{matrix}{{\Psi\rangle} = \left( {\underset{\underset{({2^{n} - t})}{}}{{a_{0},\ldots \quad,}\quad}\quad \underset{\underset{t}{}}{a_{1},\ldots \quad,}} \right)} & (19)\end{matrix}$

In this equation, 0≦a₀<a₁ is assumed, the number of components a, isassumed to be (2^(n)−t), and the number of a₁ is assumed to be (t). Inthis case, 2≦t≦2^(n)−2 is assumed and (t) is assumed to be an evennumber. If the conditions 0<t<2^(n−2) and [(3·2^(n−2)−t)/(2^(n−2)−t)]a₀<a₁ are established, |ψ> meets S<2^(n−2) (a₀+a₁).

In this case, the difference between coefficients A₀ and A₁ can bereduced performing the inversion about average operation D and then anoperation for selectively rotating by π the phases of basic vectorshaving a negative coefficient. FIGS. 8A, 8B and 8C simply show thisoperation. FIGS. 8A, 8B and 8C show the coefficients of basic vectors of(8A)|ψ>, (8B)D|ψ>, and (8C)R(π)D|ψ>. These figures show that thedifference between coefficients decreases after the application of(R(π)D) to |ψ>. Thus, [S−2^(n−2) (a _(min)+a_(min+1))] is expected to beincreased by continuously applying (R(π)D). This will be clarified usingthe next lemma.

(Lemma 0.2)

A state given as shown below is considered.${{\Psi}\rangle} = {\left( {{\underset{\underset{2h}{}}{a_{0},\ldots \quad,}\quad \underset{\underset{2m}{}}{a_{1},\ldots \quad,}\quad a_{2{({h + m})}}},\ldots \quad,a_{N}} \right).}$

In this case, N=2^(n)−1 and 0≦a₀<a_(i)<a₁ for j=2(h+m), . . . , N. Thenumber of components a₀ is assumed to be 2h, the number of components a₁is assumed to be 2m, and h≧1, m≧1, and h+m≦2^(n−1) are assumed. S, whichis the sum of all the coefficients of |ψ>, is assumed to meet thefollowing condition.

S−2 ^(n−2)(a ₀ +a ₁)<0.  (20)

The following state is assumed to have been obtained by performing theinversion about average operation on lip>and selectively rotating by nthe phases of basic vectors having a negative coefficient.

R(π)D|ψ>=[B ₀ , . . . , B ₁ , . . . , B ₂(h+m), . . . , B _(N)],

S′ is defined as the sum of all the coefficients of R(π)D|ψ>.

1. The following relations are established. For

j=2(h+m), . . . , N(=2^(n)−1),0<B ₀ <B ₁ <B _(j) and [S′−2^(n−2)(B ₀ +B₁)]−[S−2^(n−2)(a ₀ +a ₁)]>(2h−2^(n−1))a ₀+(2^(n)−2h)a ₁>0.  (21)

2. When the following quantity is defined, $\begin{matrix}\left\{ \begin{matrix}{ɛ^{(0)} = {{\left( {{2h} - 2^{n - 1}} \right)a_{0}} + {\left( {2^{n} - {2h}} \right)a_{1}}}} \\{ɛ^{(1)} = {{\left( {{2h} - 2^{n - 1}} \right)B_{0}} + {\left( {2^{n} - {2h}} \right)B_{1}}}}\end{matrix} \right. & (22)\end{matrix}$

the following relation is established.

ε⁽¹⁾>ε⁽⁰⁾>0.

(Proof)

D|ψ> is determined as follows.

D|ψ>=[a′ ₀ , . . . ,a′ ₁ , . . . ,a′ _(2(h+m)) , . . . , a′ _(N)]

In this case, N=2^(n)−1 $\begin{matrix}\left\{ \begin{matrix}{{2^{n - 1}a_{0}^{\prime}} = {S - {2^{n - 1}a_{0}}}} & \quad \\{{2^{n - 1}a_{1}^{\prime}} = {S - {2^{n - 1}a_{1}}}} & \quad \\{{2^{n - 1}a_{j}^{\prime}} = {S - {2^{n - 1}a_{j}}}} & {\quad {{{for}\quad 2\left( {h + m} \right)} \leqq j \leqq {2^{n} - 1}}}\end{matrix} \right. & (23)\end{matrix}$

Obviously, S−2^(n−1)a₀>0. Using condition S−2^(n−2)(a₀+a₁)<0 providesrelation S−²⁻¹a_(k)<0 for ^(∀)k≠0. Thus, R(π)D|ψ> is obtained as shownbelow. $\begin{matrix}\left\{ \begin{matrix}{{2^{n - 1}B_{0}} = {S - {2^{n - 1}a_{0}}}} & \quad \\{{2^{n - 1}B_{1}} = {{- S} + {2^{n - 1}a_{1}}}} & \quad \\{{2^{n - 1}B_{j}} = {{- S} + {2^{n - 1}a_{j}}}} & {{{for}\quad 2\left( {h + m} \right)} \leqq j \leqq {2^{n} - 1}}\end{matrix} \right. & (24)\end{matrix}$

The difference between B₁ and B₀ is determined as follows.

2^(n−1)(B ₁ −B ₀)=−2[S−2^(n−2)(a ₀ +a ₁)]>0.  (25)

In this case, the assumption in Equation 20 has been used. In addition,B₁<B_(j) is obvious for j=2(h+m), . . . 2^(n)−1. Thus, 0<B₀<B₁<B_(j) isobtained for j=2(h+m), . . . , 2−1.

Based on the following relations,${S^{\prime} = {{\frac{4h}{2^{n - 1}}\left( {S - {2^{n - 1}a_{0}}} \right)} - S}},{{B_{0} + B_{1}} = {a_{1} - {a_{0}.}}}$

Δ, which denotes a change in [S−2^(n) ⁻²(a₀+a₁)] caused by the (R(π)D)operation, can be rewritten as shown below. $\begin{matrix}\begin{matrix}{\Delta = \quad {\left\lbrack {S^{\prime} - {2^{n - 2}\left( {B_{0} + B_{1}} \right)}} \right\rbrack - \left\lbrack {S - {2^{n - 2}\left( {a_{0} + a_{1}} \right)}} \right\rbrack}} \\{= \quad {{2\left( {\frac{2h}{2^{n - 1}} - 1} \right)S} - {\left( {{4h} - 2^{n - 1}} \right){a_{0}.}}}}\end{matrix} & (26)\end{matrix}$

To precisely evaluate Δ, several convenient inequalities are provided.Defining S provides the following relation. $\begin{matrix}{S = {{{2{ha}_{0}} + {2{ma}_{1}} + {\sum\limits_{k = {2{({h + m})}}}^{2^{n} - 1}a_{k}}} \geqq {{2{ha}_{0}} + {\left( {2^{n} - {2h}} \right){a_{1}.}}}}} & (27)\end{matrix}$

Assumptions (20) and (27) allow the following relation to be derived.$\begin{matrix}\begin{matrix}{0 > \quad {S - {2^{n - 2}\left( {a_{0} + a_{1}} \right)}}} \\{\geqq \quad {{2{ha}_{0}} + {\left( {2^{n} - {2h}} \right)a_{1}} - {2^{n - 2}\left( {a_{0} + a_{1}} \right)}}} \\{= \quad {{\left( {{2h} - 2^{n - 2}} \right)a_{0}} + {\left( {{3 \cdot 2^{n - 2}} - {2h}} \right){a_{1}.}}}}\end{matrix} & (28)\end{matrix}$

Then, rougher inequalities can be obtained by modifying (28).

0>2ha ₀−2^(n−2) a ₁+(3·2^(n−2)−2h)a ₁=2ha ₀+(2^(n−1)−2h)a ₁.   (29)

Since 0≦a₀<a₁, the following inequality is obtained.

2h−2^(n−1)>0.  (30)

Reviewing relations (30) and (28) allows the following inequality to beobtained.

2h>3·2^(n−2).  (31)

Then, the system is ready for strictly evaluating Δ. Relation (31)enables (27) to be assigned to Equation (26). $\begin{matrix}\begin{matrix}{\Delta \geqq \quad {{2{\left( {\frac{2h}{2^{n - 1}} - 1} \right)\left\lbrack {{2{ha}_{0}} + {\left( {2^{n} - {2h}} \right)a_{1}}} \right\rbrack}} - {\left( {{4h} - 2^{n - 1}} \right)a_{0}}}} \\{{{\left. {= \quad \left. \left( {1/2^{n - 1}} \right. \right\}} \right)\left\lbrack {{8h^{2}} - {8{h \cdot 2^{n - 1}}} + 2^{2{({n - 1})}}} \right\rbrack}a_{0}} + \left( {1/2^{n - 1}} \right)} \\{\quad {\left\lbrack {{{- 8}h^{2}} + {6{h \cdot 2^{n}}} - 2^{2n}} \right\rbrack a_{1}}} \\{{\left. {= \quad {{\left. {\left( {1/2^{n - 1}} \right)\left\lbrack \left. \left( {{4h} - {3 \cdot 2^{n - 1}}} \right. \right\} \right.} \right)\left( {{2h} - 2^{n - 2}} \right)} - 2^{{2n} - 3}}} \right\rbrack a_{0}} +} \\{{\quad \left. {\left. \left. {\left( {1/2^{n - 1}} \right)\left\lbrack {\left( {{{- 4}h} + {3 \cdot 2^{n - 1}}} \right)\left( {{2h} - {3 \cdot 2^{n - 2}}} \right.} \right.} \right\} \right) + 2^{{2n} - 3}} \right\rbrack}a_{1}} \\{= \quad {{\left( {1/2^{n - 1}} \right){\left( {{4h} - {3 \cdot 2^{n - 1}}} \right)\left\lbrack {{\left( {{2h} - 2^{n - 2}} \right)a_{0}} + {\left( {{3 \cdot 2^{n - 2}} - {2h}} \right)a_{1}}} \right\rbrack}} +}} \\{\quad {2^{n - 2}{\left( {a_{1} - a_{2}} \right).}}}\end{matrix} & (32)\end{matrix}$

Relation (31) indicates that 3·2^(n−2)<2h<2^(n). Thus, relation0<(4h−3·2^(n−1))<2^(n−1) is derived.

Equation (28) enables Δ to be evaluated as follows. $\begin{matrix}\begin{matrix}{\Delta > \quad {\left\lbrack {{\left( {{2h} - 2^{n - 2}} \right)a_{0}} + {\left( {{3 \cdot 2^{n - 2}} - {2h}} \right)a_{1}}} \right\rbrack + {2^{n - 2}\left( {a_{1} - a_{0}} \right)}}} \\{= \quad {{\left( {{2h} - 2^{n - 1}} \right)a_{0}} + {\left( {2^{n} - {2h}} \right)a_{1}}}} \\{> \quad 0.}\end{matrix} & (33)\end{matrix}$

Thus, the first result has been derived.

Definitions (22), (24), (27), and (28) enable the difference betweenε⁽⁰⁾ and ε⁽¹⁾. $\begin{matrix}\begin{matrix}{{ɛ^{(1)} - ɛ^{(0)}} = \quad {{\left( {{2h} - 2^{n - 1}} \right)\left( {B_{0} - a_{0}} \right)} + {\left( {2^{n} - {2h}} \right)\left( {B_{1} - a_{1}} \right)}}} \\{= \quad {\frac{1}{2^{n - 1}}\left\lbrack {{\left( {{4h} - {3 \cdot 2^{n - 1}}} \right)S} - {2^{\bigwedge}\left\{ n \right\} {a_{0}\left( {{2h} - {2^{\bigwedge}\left\{ {n - 1} \right\}}} \right)}}} \right\rbrack}} \\{\geqq \quad {\frac{1}{2^{n - 1}}\left\{ {{\left( {{4h} - {3 \cdot 2^{n - 1}}} \right)\left\lbrack {{2{ha}_{0}} + {\left( {2^{n} - {2h}} \right)a_{1}}} \right\rbrack} -} \right.}} \\{\quad \left. {\left( {{2h} - 2^{n - 1}} \right)2^{n}a_{0}} \right\}} \\{= \quad {{\frac{1}{2^{n - 1}}\left\lbrack {{\left( {{2h} - {3 \cdot 2^{n - 2}}} \right)a_{1}} - {\left( {{2h} - 2^{n - 2}} \right)a_{0}}} \right\rbrack}\left( {2^{n} - {2h}} \right)}} \\{> \quad 0.}\end{matrix} & (34)\end{matrix}$

Thus, the second result has been derived.

According to lemma 0.2, the (R(π)D) transformation can be continuouslyapplied to constantly make [S−2^(n−2)(a₀+a₁)] positive or zero. State|ψ⁽⁰⁾> specified by coefficients {a₀, a₁, a_(2(h+m)), a_(2(h+m)+1), . ..,a 2_(n−1)} is assumed to meet S−2^(n−2) (a₀+a₁)<0. State|ψ⁽¹⁾>specified by coefficients {B₀, B₁, B_(2(l+m)), B_(2(l+m)+1), . . ., B_(2n−1)) is assumed to have been obtained by applying (R(π)D) to|ψ⁽⁰⁾>. Lemma 0.2.1 provides the following relation:

[S−2^(n−2)(B ₀ +B ₁ ]−[S−2^(n−2)(a ₀ +a ₁)]>ε⁽⁰⁾>0,  (35)

where ε⁽⁰⁾ is a quantity defined by (22).

Next, |ψ⁽¹⁾> is assumed to meet S′−2^(n−2) (B₀+B₁)<0. |ψ⁽²⁾> specifiedby {B₀ ⁽²), B₁(2), B_(2(h+m)) ⁽²⁾, B_(2(h+m)+1,) ⁽²⁾, . . ., B_(N) ⁽²⁾}is assumed to have been obtained after applying (R(π)D) to |ψ⁽¹⁾>. Lemma0.2.2 provides the following relation.

[S′ ⁽²⁾−2^(n−2)(B₀ ⁽²⁾ +Bb ₁ ⁽²⁾)]−[S′−2^(n−2)(B ₀ +B₁)]>ε⁽¹⁾>ε⁽⁰⁾>0.  (36)

Accordingly, if [S′^((j))−2^(n−2)(B₀ ^((j))+B₁ ^((j)))]<0 is met,[S′^((j+1))−2^(n−2) (B₀ ^((j+1))+B₁ ^((j+1)))] increases by at leastε⁽⁰⁾ (>0). (j) indicates the number of times that (R(π)D) has beenapplied. ε⁽⁰⁾ is defined by {a₀, a₁} and h.

Thus, e⁽⁰⁾ is an established positive value. The repeated application of(R(π)D) ensues that [S′−2^(n−2) (B₀+B₁)] is positive or zero.

If [S′^((j))−2^(n−2) (B₀ ^((j))+B₁ ^((j))] becomes positive or zero, the(R′DR) contraction method is started again. Using the (R′DR) contractionmethod and the (R(π)D) repetition operation, |ψ_(n)> can be constantlytransformed into a flat superposition.

Then, it is considered how many times (R(π)D) should be applied to |ψ>in order to meet S′^((j))}−2^(n−2) (B₀ ^((j))+B₁ ^((j))>0. To evaluatethis, the following state is assumed.${{\Psi}\rangle} = \left( {{\underset{\underset{t_{0}}{}}{a_{0},\ldots \quad,}\quad \underset{\underset{t_{1}}{}}{a_{1},\ldots \quad,}\quad a_{2}},\ldots \quad,\underset{\underset{t_{M}}{}}{a_{M},\ldots}} \right)$

In this case ${{\sum\limits_{k = 0}^{M}t_{k}} = 2^{n}},$

0≦a₀<a₁<a_(k) for k=2, . . . M, and S−2^(n−2) (a₀+a₁)<0. Then, a₀, a₁, .. . a_(M) are expressed as shown below.${a_{0} = \frac{\sin \quad \alpha_{0}}{\sqrt{t_{0}}}},{a_{1} = \frac{\cos \quad \alpha_{0}\sin \quad \alpha}{\sqrt{t_{1}}}},\ldots \quad,{a_{M} = {\frac{\cos \quad \alpha_{0}\cos \quad \alpha_{1}\quad \cdots \quad \cos \quad \alpha_{M - 1}}{\sqrt{t_{M}}}.}}$

ε⁽⁰⁾ and the order of [S−2⁻² (a₀+a₁)] can be evaluated as follows.

ε⁽⁰⁾=(2^(n−1) −t)a ₀ +ta ₁>2^(n−1) a ₀>∘(2^((n/2)−1)),  (37)

S−2 ^(n−2)(a ₀ +a ₁)>2^(n) a ₀−2^(n−2)(a ₀ +a ₁)=3·2^(n−2) a ₀−2^(n−2) a₁>−∘(2^(n−2).  (38)

Thus, the number of times T that (R(π)D) is applied is given as shownbelow.$T \sim {- \frac{S - {2^{n - 2}\left( {a_{0} + a_{1}} \right)}}{ɛ^{(0)}}} \sim {{O\left( 2^{{({n/2})} - 1} \right)}.}$

With Equation (24), {B_(i)} can be computed within about 0(n) stepsusing a conventional computer. This is because the number of differentcoefficients in {B_(i)} is (└n/2┘+1). Next, the simplest example of the(R(π)D) repetition operation expressed by Equation (19) will be examinedaccurately.

[Simple Example of the (R(π)D) Repetition Method]

A simple example of the (R(π)D) repetition operation expressed byEquation (19) is considered, and evaluations are conducted to determinehow many times (R(π)D) should be applied to make [S′−2^(n−2)(A₀+A₁)]positive or zero.

(R(π)D) is applied to |ψ> given by Equation (19) to obtain R(π)D|ψ>=[B₀,. . . , B₁, . . .]. In this case, the following equations areestablished. $\begin{matrix}\left\{ \begin{matrix}{{2^{n - 1}B_{0}} = {{S - {2^{n - 1}a_{0}}} = {{\left( {2^{n - 1} - t} \right)a_{0}} + {ta}_{1}}}} \\{{2^{n - 1}B_{1}} = {{{- S} + {2^{n - 1}a_{1}}} = {{{- \left( {2^{n} - t} \right)}a_{0}} + {\left( {2^{n - 1} - t} \right)a_{1}}}}}\end{matrix} \right. & (39)\end{matrix}$

(t) is expressed as shown below. $\begin{matrix}{{{\sin^{2}\theta} = \frac{t}{2^{n}}},\left( {{\cos^{2}\theta} = \frac{2^{n} - t}{2^{n}}} \right),} & (40)\end{matrix}$

where 0<θ<(n/2). In addition, {a₀, a₁} meets the following equations:$\begin{matrix}{{a_{0} = \frac{\sin \quad \alpha}{\sqrt{2^{n} - t}}},{a_{1} = \frac{\cos \quad \alpha}{\sqrt{t}}},} & (41)\end{matrix}$

where 0≦α<(π/2) (in handling this model, the following document wasreferenced: M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight boundson quantum searching,” LANL quantum physics archive quant-ph/9605034).

Using Equations (39), (40), and (41), {B₀, B₁} can be expressed as shownbelow. $\begin{matrix}\left\{ \begin{matrix}{B_{0} = {\left( {1/\sqrt{2^{n}}} \right)\left\lbrack {{\cos \quad 2{\theta \left( {\sin \quad {\alpha/\cos}\quad \theta} \right)}} + {2\sin \quad \theta \quad \cos \quad \alpha}} \right\rbrack}} \\{\quad {= {{\sin \left( {\alpha + {2\theta}} \right)}/\sqrt{2^{n} - t}}}} \\{B_{1} = {\left( {1/\sqrt{2^{n}}} \right)\left\lbrack {{{- 2}\cos \quad {\theta sin}\quad \alpha}\quad + {\cos \quad 2\theta \quad \left( {\cos \quad {\alpha/\sin}\quad \theta} \right)}} \right\rbrack}} \\{\quad {= {{\cos \left( {\alpha + {2\theta}} \right)}/\sqrt{t}}}}\end{matrix} \right. & (42)\end{matrix}$

The coefficients of a state obtained after (j) times of (R(π)D)application are represented as B₀ ^((j) and B) ₁ ^((j)). This can bewritten as follows: $\begin{matrix}{{B_{0}^{(j)} = {\frac{1}{\sqrt{2^{n} - t}}{\sin \left( {\alpha + {2j\quad \theta}} \right)}}},{B_{1}^{(j)} = {\frac{1}{\sqrt{t}}{{\cos \left( {\alpha + {2j\quad \theta}} \right)}.}}}} & (43)\end{matrix}$

where B₀ ⁽⁰⁾=a₀ and B₁ ⁽⁰⁾=a₁.

When S^((j))(2^(n)−t) B₀ ^((j))+tB₁ ^((j)), the following is derived.$\begin{matrix}\begin{matrix}{{S^{(j)} - {2^{n - 2}\left( {B_{0}^{(j)} + B_{1}^{(j)}} \right)}} = \quad {{\left( {{3 \cdot 2^{n - 2}} - t} \right)B_{0}^{(j)}} + {\left( {t - 2^{n - 2}} \right)B_{1}^{(j)}}}} \\{= \quad {\sqrt{2^{n}}\left\{ {{\sin \left\lbrack {\alpha + {\left( {{2j} + 1} \right)\theta}} \right\rbrack} -} \right.}} \\\left. \quad {\frac{1}{2\quad \sin \quad 2\quad \theta}{\cos \left\lbrack {\alpha + {\left( {{2j} - 1} \right)\quad \theta}} \right\rbrack}} \right\} \\{= \quad {{- \frac{\sqrt{2^{n - 2}}}{\sin \quad 2\quad \theta}}{F^{(j)}.}}}\end{matrix} & (44)\end{matrix}$

In this case, the following equation is established.

F ^((j))=cos[α+(2j+3)θ]  (45)

Since 0<θ<(π/2) and sin 2θ>0, whether [S^((j))−2^(n−2)(B₀ ^((j))+B₁^((j))] is zero or positive depends on the sign of F^((j)).

Based on 0≦α< (π/2), if (2j+3)θ=(π/2), then constantly F^((j))≦0, thatis, S^((j))−2^(n−2) (B₀ ^((j))+B₁ ^((j)))≧0. When j_(MAX) is specifiedas shown below, the number of times that (R(π)D) is repeated isprevented from exceeding j_(MAX).$j_{MAX} = {\frac{1}{2\quad \theta}{\left( {\frac{\pi}{2} - {3\quad \theta}} \right).}}$

On the other hand, Equation (40) enables e to be written as${\sin \quad \theta} = {\sqrt{\frac{t}{2^{n}}}.}$

The minimum value of (t) is 2. If t−0(1) and (n) is sufficiently large,the following relation is obtained.${\sin \quad \theta} \sim \theta \sim {\sqrt{\frac{t}{2^{n}}}.}$

Setting such a limit provides the following relation.$j_{MAX} \sim {\frac{\pi}{4}\sqrt{\frac{2^{n}}{t}}} \sim {{O\left( 2^{n/2} \right)}.}$

This indicates that [S^((j))−2^(n−2) (B₀ ^((j))+B₁ ^((j))] can be madepositive or zero by repeating the (R(π)D) operation at most 0(2^(n/2))times.

[Configuring a Partly Entangled State Given by a Function with an EvenNumber of Collisions]

The partly entangled state having a high symmetry has been discussed.The method described herein, however, is applicable to El partlyentangled state that is more generalized.

For the time being, the following function will be considered:

f:A={0,1}^(n) →B={0,1}^(m)  (46)

where 0≦m≦n−1.

∀ε{0,1}^(m)

The number of xε{0, 1}^(n) meeting f(x)=y for ∀yε{0, 1}^(m) is an evennumber including zero. FIG. 9 shows a mapping caused by (f). The numberof all the collisions in (f) is even.

The elements of images that are mapped from {0, 1}^(n) by (f) arewritten as {β₀, β₁, β₂, . . . , β_(M)} where 0≦M≦2^(m−)1. Then, thefollowing equation is assumed.

m=┌log₂(M+1)┐

This equation serves to save the number of qubits of the secondregister.

In addition, the elements of A={0, 1}^(n) that are mapped toβ_(k)(0≦k≦M) by (f) is written as {α(k, 1), α(k, 2), . . . , α(k,h_(k))}. h_(k) is an even number. h_(k) is the number of elements in Athat may collide when β_(k)εB. Then, the following state is assumed:$\begin{matrix}{{{{\Psi_{n}}\rangle} = {\sum\limits_{k = 0}^{M}\quad {\sum\limits_{h = 1}^{h_{k}}\quad {a_{k}{{\alpha \left( {k,h} \right)}\rangle}}}}},} & (47)\end{matrix}$

where {a_(k)}is a real number. By providing quantum gates U_(f) forcomputing f(x) and using the method described above, such |ψ_(n)> can beconfigured. In particular, for a partly entangled state having a highsymmetry, function (f) is written as shown below.

f(i)=(number of “1” bits in the binary expression of (i)) However,(n−f(i)) is considered to be identical to f(i), and when:

m=┌log ₂(n+1)┐

A={0, 1}^(n),

B={0, 1}^(m).

Then, the method described above will be explained again in brief. Sincethe method for transforming |ψ_(n)> into a flat superposition is simplerthan the method for transforming the flat superposition |ψ_(n)>, theformer method is described in short. To actually operate quantum gates,the inverse transformation of the following method may be used andcorresponds to the inversion of the direction of the time of a Feynmandiagram of a network.

1. All basic vectors having a negative coefficient are selectivelyrotated by π. If the state of the register is equal to a flatsuperposition, the operation is terminated. Otherwise, the processproceeds to step 2.

2. a_(min) is the value of the minimum coefficient of basic vectors, anda_(min+1) is the second smallest coefficient. a_(min) and a_(min+1) arechecked for the fulfillment of the sufficient condition for the (RIDR)contraction method. If they meet the sufficient condition, the processpasses to step 1. Otherwise, the process advances to step 3.

3. The process executes the (R(π)D) transformation and then passes tostep 1.

[Example of a Specific, Quantum Gate Network and Evaluations of theComputation Time]

A network of quantum gates that constitute |ψ_(n)> defined by Equation(47) will be considered. Two registers and function (f) given byEquation (46) are provided. $\begin{matrix}{{{{{{{{{x}\rangle} \otimes {y}}\rangle}\overset{f}{}{x}}\rangle} \otimes {{y \oplus {f(x)}}}}\rangle},} & (48)\end{matrix}$

The first register is composed of (n) qubits, and the second register iscomposed of (m) qubits. The second register is used as a control sectionfor selective rotations.

To configure |ψ_(n)> from a flat superposition, the (R,DR) contractionoperation must be performed M times. Thus, the (DR(π)) transformation iscarried out M×0 (2^(n/)2) times at maximum. If M is not in a polynominalorder of (n), the method described herein is not effective. Then,function (f), the selective rotation, and a network for D will bespecifically discussed.

(First Embodiment)

A quantum gate network for function (f) is considered. FIG. 10 shows aFeynman diagram of function (f). (f) is expressed as a quantum controlgate that carries out a unitary transformation in the second registerusing a value from the first register.

If the (f) control gate can be composed of an elementary quantum gate ina poly(n) order (a polynominal order of (n)), the method described inthis embodiment can operate efficiently. In particular, for a functiongiven by f(i)=s(i), the (f) control gate can be composed of poly(n)steps.

A design of a specific quantum network for a control gate for f(i) =s(i)is shown below. The first and second registers are written as shownbelow.

|X_(n), X_(n−1), . . . X₂, X₁>{circle around (x)}|S>, The quantum gatenetwork can be expressed as the following program (To write such aprogram, R. Cleve and D. P. DiVincenzo, ¹“Schumacher's quantum datacompression as a quantum computation,” Phys. Rev. A 54, 2636 (1996) hasbeen referenced).

Program QUBIT-ADDER1

for k=1 to n do

S←(S+X_(k))mod 2^(m).

In the above description, X_(i) means the ith qubit of the firstregister. In addition, S represents the value of the second register.(m) represents the number of qubits in the second register andm=┌log₂(n+1)].

Then, auxiliary qubits {C₁, . . . , C_(m−1)} are introduced. As shown inFIG. 11, Cj represents a carry due to the addition of the (j−1)th bit.The qubits of the second register are expressed as {S₀, S₁, . . .S_(m−1)}. Thus, a program for addition of X_(k) in program QUBIT-ADDER1can be written as follows.

Program QUBIT-ADDER2

quantum registers:

X_(k); a qubit register

S₀, S₁, . . . , S_(m−1), ; qubit registers

C₁, C₂, . . . , C_(m−1); auxiliary qubit registers

(initialized and finalized to 0)

C ₁ →C ₁⊕(S ₀ ΛX _(i))

for j=2 to m−1 do

C _(j) ←C _(j)⊕(C _(j−1) ΛS _(j−1))

for j=m−1 down to 2 do

S _(j) ←S _(j) ⊕C _(j)

C _(j) ←C _(j)⊕(C _(j−1) ΛS _(j−1))

S ₁ ←S ₁ ⊕C ₁

C ₁ ←C ₁⊕(S ₀ ΛX _(i))

S ₀ ←S ₀ ⊕X _(i)

FIG. 12 is a Feynman diagram of this program for m=6. ProgramQUBIT-ADDER1 can be configured by repeating for each X_(i) (i =1, . . ., n) the network shown in FIG. 12, as shown in FIG. 13.

Next, evaluations are made as to how many elementary quantum gates arerequired to configure QUBIT-ADDERI. In FIG. 12, 2(m−1) Toffoli gates and(m) controlled-NOT gates are used to configure QUBIT-ADDER2.

The total number of steps required for QUBIT-ADDER1 is n(3m−2) due to(n) times of repetition of QUBIT-ADDER2. According to this embodiment,any unitary transformation ∀UEU(2) for the Toffoli gate, controlled-NOTgate, or 1-qubit is considered to be one unit.

(Second Embodiment)

Another important quantum network used in the method described herein isa selective rotation. For example, in executing the (R,DR) contractionmethod, the phases of the two states of the second register are rotatedby (±θ).

FIG. 14 shows a quantum gate network with a selective rotation in thesecond register. This network uses two controlled^(m)-R_(Z)(α) gates. Acontrolled^(m)-U gate having m-qubit control section and one-qubittarget section is written as Λ_(m)(U). In this case, ∀UEU(2) isestablished.

Λ_(m)(U) operates as described below. If all the (m) qubits of thecontrol section is equal to |1>, Λ_(m)(U) applies a U transformation tothe target qubit. If the m-qubit control section is not in the |1 . . .1> state, Λm(U) does nothing to the target qubit. For example, a Toffoligate is represented as Λ₂(σ_(x)) and a controlled-NOT gate isrepresented as Λ₁(σ_(x)) A represents an arbitrary U(2) gate. R_(Z)(α)is one form of U(2) transformation and is given as shown below.${R_{z}(\alpha)} = {{\exp \left( {\frac{}{2}\quad \alpha \quad \sigma_{z}} \right)} = \begin{bmatrix}{\exp \quad \left( {\quad {\alpha/2}} \right)} & 0 \\0 & {\exp \left( {{- }\quad {\alpha/2}} \right)}\end{bmatrix}}$

If auxiliary qubit is set to 0>, Λ_(m)(R_(Z)(α)) emits unique valueexp(iα/2) only when the second register is in the |1 . . . 1>. Such amethod is called “kicked back” (see R. Cleve, A. Ekeart, C.Macchiavello, and M. Mosca, “Quantum Algorithms Revisited, LANL quantumphysics archive quant-ph/9708016).

In FIG. 14, the shaded boxes represent NOT gates or identitytransformations. The NOT-gate is given by σ_(x). Basic vectors to berotated are determined depending on whether the shaded box is a NOT-gateor an identity transformation.

For (m≧6), Λ_(m)(R_(Z)(α)) can be composed of 16(m−4)Λ₂(σ_(x)) gates, 4Λ_(x)(σ_(x)) gates, and 4 Λ₀ gates. This will be shown below (reference:Barenco, C. H. Bennett, R. Cleve, D. P. Divincenzo, N. Margolus, P.Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates forquantum computation,” Phys. Rev. A52, 3457 (1995)).

Thus, for (m≧6), Λ_(m)(R_(Z)(α)) is composed of at most 8(2m−7) quantumelementary gates. FIG. 14 shows that the selective rotation in thesecond register is composed of at most the following number of steps.

3m+2·8(2m−7)=7(5m−16)˜0(m).

Furthermore, if (n) is an even number and η=η−η=n/2, the use of firstqubit X1 of the first register enables the phase to be rotated by θ forhalf the elementary vectors with s(k)=η while enabling the phase to berotated by (−θ) for the remaining basic vectors. FIG. 15 shows a networkfor this operation. In configuring |ψ_(n)>, the selective rotation inthe second register can be executed in about 0(log₂n) steps.

(Third Embodiment)

Finally, a network for D is considered. Before applying D to the firstregister, the second register must be initialized to |0 . . . 0>. Thus,in executing the (R′DR) contraction method, (R′U_(f)DU_(f) ⁻¹R) must beapplied to the register where U_(f) is an unitary transformation offunction (f) defined by Equation (48).

D is known to be decomposable as follows (see L. K. Grover, “A fastquantum mechanical algorithm for database search,” LANL quantum physicsarchive quant-ph/9605043 and L. K. Grover, “Quantum Mechanics Helps inSearching for a Needle in a Haystack,” Phys. Rev. Lett. 79, 325 (1997)).

D=H ^((n)) RH ^((n))

but

H ^((n)) =H{circle around (x)} . . . {circle around (x)}H

means a Hadamard transformation for n-qubits, and R represents aselective rt rotation for |0 . . . 0> in the first register.

FIG. 16 shows a network for D. The network consists of 4n Λ₀ gates andone Λ_(n)(R_(Z)(2π)) gates, requiring 4(5n−14) elementary operations. Dis composed of 0(n) steps.

(Fourth Embodiment)

The number of elementary operations required to configure |ψ_(n)> from aflat superposition is evaluated in this embodiment. If M (number ofcomponents of an image in mapping (f) defined by Equation (46)) is aboutpoly(n) and function U, defined by Equation (48) can be composed ofabout poly(n) elementary gates, repetitive operations of (R(π)D)definitely occupy most of the steps.

A network for subroutine (R(π)D) is given in FIG. 17. Although thenumber of steps required to configure function Uf depends on (f), (firstembodiment) has already shown that about 0(nlog₂n) steps are required toconfigure, for example, |ψ_(n)>. In addition, (third embodiment) hasalready shown that 4(5n−14) steps are required for D.

FIG. 18 shows a Feynman diagram of a selective π rotation in the (R(π)D)operation. In this case, R(π) is an operation for inverting the signs ofbasic vectors having a negative coefficient, so at most P=└M/2┘ basicvectors of the second register are rotated.

Thus, if: P=└M/2┘

Q┌=log₂(M+1)┐

then the selective rotation is composed of (P+1)·QΛ₀ gates and P ΛQ(R_(Z)(2π)) gates. R(π) can be carried out in 0(Mlog₂M) steps.

A possible evaluation is that the (R(π)D) transformation requires0(nlog₂n) steps in order to configure |ψ_(n)>. The number of times that(R(π)D) is repeated between (R′{η}DR{η}) contraction operation is atmost about 0(2^(n/2)). If the (R(π)D) repetition operation is performedeach time the (R′{η}DR{η}) contraction operation is performed, the(R(π)D) repetition operation is performed └n/2┘ times. Thus, in total,0(n²log₂n)×2^(n/2)) steps are required.

In general, an arbitrary transformation U(εU(2^(n))) is known to becomposable of at most 0(n³2^(2n)) elementary gates (see A. Barenco, C.H. Bennett, R. Cleve, D. P. Divincenzo, N. Margolus, P. Shor, T.Sleator, J. Smolin, and H. Weinfurter, “Elementary gates for quantumcomputation,” Phys. Rev. A52, 3457 (1995)). Thus, the method describedin this embodiment reduces the number of elementary quantum gatescompared to conventional cases.

In addition, in configuring |ψ_(n)>, a quantum gate network can becomposed of only the (R′{η}DR{η}) contraction method without a (R(π)D)repetition operation, depending on the value of {a_(n)}. A network forthis case is given in FIG. 19.

First, Hadamard transformation H is applied independently to (n) qubitsof the first register. Next, a block U_(add) is operated that computesthe total number of bits in the first register that are set to ‘1’.Subsequently, a set of three operations including a selective rotation,an inversion about average operation U_(add)D U_(add) ⁻¹, and aselective operation are repeated (p) times.

Finally, to return all the qubits of the second register to initialstate |0>, U_(add) ⁻¹ is operated. U_(add) ⁻¹, which is a network blockfor carrying out the inverse transformation of U_(add), operates theFeynman diaphragms in FIGS. 12 and 13 leftward, that is, in thedirection opposite to the time. Thus;, the total amount of computationsis expressed as shown below.

n+2×n┌log₂(n+1)┐−2)+└n/2┘{2×7(5┌log₂(n+1)┐−16)+4(5n−14)+2×n(3┌log₂(n+1)┐−2)}˜3n²log₂ n

Furthermore, according to the method described herein, beforeconstructing a quantum gate network, a conventional computer must beused to determine the phase rotation parameter and the order of basicvectors with their phase selectively rotated. In this case, the numberof steps required for conventional computations can also be limited toabout 0((n²log₂n)×2^(n/2)).

(Fifth Embodiment)

This embodiment considers how a Λ_(n)(R_(Z)(α)) gate is configured usingelementary quantum gates. The cited document (A. Barenco, C. H. Bennett,R. Cleve, D. P. Divincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin,and H. Weinfurter, “Elementary gates for quantum computation,” Phys.Rev. A52, 3457 (1995)) discloses various approaches for configuringΛ_(n)(U) using elementary quantum gates for (∀UEU(2)).

First, the following relations are noted.

R _(Z)(α/2)σ_(x) R _(Z)(−α/2)σ_(x) =R _(Z)(α)

R _(Z)(α/2)R _(Z)(−α/2)=I

Thus, as shown in FIG. 20, Λ_(n)(R_(Z)(α)) can be decomposed intoΛ₁(R_(Z)(α/2)), Λ₁(R_(Z)(−α/2)), and two Λ_(n−1)(σ_(x)). Furthermore,Λ₁(R_(Z)(β)) can be decomposed into R_(Z)(β/2), R_(Z)(−β/2), and twocontrolled-NOT gates.

Then, for the time being, how a Λ_(n−1)(σ_(x)) gate is configured on a(n+1)-qubit network will be discussed. It should be particularly notedthat one qubit that has not been used by Λ_(n−1)(σ_(x)) remains on thenetwork.

For n=4, Λ₃(σ_(x)) can be decomposed into four Toffoli gates. For n=5,Λ₄(σ_(x)) can be decomposed into two Toffoli gates and two Λ₃(o() gates,so Λ₄(σ_(x)) can be decomposed into 10 Toffoli gates. For n=6, Λ₅(σ_(x))can be decomposed into four Λ₃(σ_(x)) gates, as is apparent from FIG.22. This means that Λ₅(σ_(x)) can be decomposed into 16 Toffoli gates.

A case of n≧7 is considered as follows. A Λ_(n−1)(σ_(x)) gate can bedecomposed into two Λ_(m1)(σ_(x)) gates and two Λ_(m2)(σ_(x)) gates. Inthis case,

m ₁┌(n+1)/2┐

m₂=n−m₁. FIG. 23 shows a case of n=8, m₁=5, and m₂=3.

Such a decomposition provides m₂ unused qubits for Λm₂(σ_(x)) whileproviding ml unused qubits for Λm₁(σ_(x)). The following relation isobtained for both Λm₁(σ^(x)) and Λm₂(σ_(x)).

(number of qubits in the control section)−(number of qubitsunused)≦2.  (49)

As regards this, the following fact is known from the cited document (A.Barenco, C. H. Bennett, R. Cleve, D. P. Divincenzo, N. Margolus, P.Shor, T. Sleator, J. Smolin, and H. Weinfurter, “Elementary gates forquantum computation,” Phys. Rev. A52, 3457 (1995)).

For n≧4, if:

mε{3, . . . , ┌n+1)/21┐}

on a (n+1)-qubit network (that is, the relation in Inequity (49) isestablished), the Λ_(m)(σ_(x)) gate can be decomposed into 4(m−2)Toffoli gates. FIG. 24 shows a case of n=8 and m=5.

Accordingly, for n≧7, on the (n+1)-qubit network, Λ_(n−1)(σ_(x)) can bedecomposed into 8(n−4) Toffoli gates based on the following equation.

2·4(m ₁−2)+2·4(m₂−2)=8(n−4 ).

FIGS. 25 and 26 show the number of elementary quantum gates required toconfigure Λ_(n−1)(σ_(x)) and Λ_(n)(R_(Z)(α)).

(Sixth Embodiment)

The method and apparatus for configuring an entangled state as describedabove can be implemented by simply providing an arbitrary unitarytransformation of one qubit and controllesd-NOT gates between twoqubits.

The cited document (J. I. Cirac and P. Zoller, “Quantum Computationswith Cold Trapped Ions,” Phys. Rev. Lett. 74), 4091 (19955)) discusses amethod for implementing controlled-NOT gates using a method called “ColdTrapped Ions. ” In this case, (n) ions are linearly captured, and thebasic and first excited states of each ion are considered to be {|0>,|1>} of a qubit. In addition, quantum gates are operated by externallyirradiating each ion with laser beams.

Linearly captured ions are engaged in Coulomb interactions, so that eachion vibrates around its balanced point. When this vibration mode isquantized, the ions become phonons, which can then be used as auxiliaryqubits. The above document ingeniously uses these phonons to implementcontrolled-NOT gates. Such a system can implement the method andapparatus for configuring an entangled state as described in the aboveembodiments.

By configuring |ψ_(n)> using (n) captured ions and carrying out theRamsey spectroscopy according to the procedure disclosed in the citeddocument (S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M.B. Plenio and J. I. Cirac, “Improvement of Frequency Standards withQuantum Entanglement,” Phys. Rev. Lett. 79, 3865 (1997)), the differencein energy level between the basic and first excited states of ions canbe measured with accuracy beyond the shot noise limit.

The method and apparatus for configuring an entangled state as describedin the above embodiments reduces the total number of elementary quantumgates required, that is, the time required for operations compared tothe configuration of general unitary transformations. As a result, thedesired state ||ψ_(n)> can be configured in a time shorter than the timescale in which a decoherence may occur in linearly captured ions.

(Seventh Embodiment)

The cited document (C. H. Bennett, C. A. Fuchs, and J. A. Smolin,“Entanglament-Enhanced Classical Communication on a Noisy QuantumChannel,” Quantum Communication, Computing, and Measurement, edited byHirota et al., Plenum Press, New York, p.79 (1997)) discusses a methodfor transferring through a noisy quantum channel, classical binaryinformation consisting of “0” and “1”

This document discloses that if two channels called “two-Pauli channels”and containing quantum noise are used, a partly entangled state of twoqubits is optimal for transmitting classical information.

Thus, the method and apparatus for configuring an entangled state asdescribed in the above embodiments is applicable to the field of quantumcommunication.

For example, as shown in FIG. 27, a transmitter A uses a quantum gatenetwork based on the method described in the embodiments in order toconfigure a target entangled state, and transmits it through twocircuits with noise. A recipient B detects this to receive classicalbinary information.

According to the above embodiments, if a quantum mechanical stateconsisting of a plurality of two-level systems is expressed by asuperposition of orthonormal bases in which each two-level systemassumes a basic or an excited state, then simple operations can be usedto configure a desired partly-entangled quantum mechanical state inwhich the coefficients of the bases are all real numbers.

In this case, the desired quantum mechanical state can be configuredusing a small number of steps.

In addition, only a small amount of computations are required todetermine the order and parameters of the operations.

In addition, by using the cold trapped ions method to provided ions asthe two-level systems, the difference in energy level between the basicand first excited states of ions can be precisely measured based on theRamsey spectroscopy.

In addition, with the quantum mechanical state configured as describedabove, transmitted information can encoded so that the quantummechanical state is communicated as a signal on a quantum communicationcircuit. In this case, information can be encoded into a state thatresists noise.

Although the present invention has been described in its preferred formwith a certain degree of particularity, many apparently widely differentembodiments of the invention can be made without departing from thespirit and scope thereof. It is to be understood that the invention isnot limited to the specific embodiments thereof except as defined in theappended claims.

What is claimed is:
 1. A stare configuring method for configuring adesired partly-entangled quantum mechanical state, expressed by asuperposition of orthonormal bases in which each two-level systemassumes a basic or an excited state and the coefficients of the basesare all real numbers, by performing an operation that is a combinationof a selective rotation operation and an inversion about averageoperation.
 2. The state configuring method according to claim 1, whereinsaid desired quantum mechanical state is defined by a function thatcauses an even number of collisions where the even number includes zero.3. The state configuring method according to claim 2, comprising:providing a first and a second registers for configuring a quantummechanical state; setting a parameter for a phase rotation; writing tosaid second register the value of said predetermined function output ifthe value of said first register is input, selectively rotating thephase according to the value of the second register and based on saidparameter for a predetermined state in said superposition in said firstregister, and subsequently providing as one unit, three continuousoperations including a selective rotation operation for returning thevalue of said second register to its initial value, an inversion aboutaverage operation that varies depending on the value of the firstregister, and a selective rotation operation for selectively rotatingthe phase again relative to said predetermined state; and repeating saidoperation unit a predetermined times to configure the desired quantummechanical state.
 4. The state configuring method according to claim 1,wherein an operation comprising a combination of said selective rotationoperation and said inversion about average operation is performed on aflat superposition to configure said desired quantum mechanical state.5. The state configuring method according to claim 4, wherein a Hadamardtransformation is applied to an initial state to obtain said flatsuperposition.
 6. The state configuring method according to claim 1,wherein said operation comprising a combination of said selectiverotation operation and said inversion about average operation is anoperation that is a combination of a series of operations forsequentially performing a selective rotation, an inversion about averageoperation, and a selective rotation; and an operation for repeating aselective n rotation and an inversion about average operation.
 7. Thestate configuring method according to claim 4, wherein in said selectiverotation for transforming said flat superstition into said desiredquantum mechanical state, the phase rotation parameter and the order inwhich the bases are rotated are determined from the procedure of atransformation inverse to said transformation.
 8. The state configuringmethod according to claim 7, wherein said procedure of inversetransformation is determined by sequentially performing operations formaking two different coefficients the same using a selective rotationoperation, an inversion about average operation, and a selectiverotation operation.
 9. A state configuring method according to claim 8,comprising: rotating the phases of bases with a negative coefficient byπ to invert their sign in order to make the coefficients of all thebases positive or zero; and sequentially performing said operations formaking two different coefficients, on the basis having the minimumcoefficient and the basis having the second smallest coefficient. 10.The state configuring method according to claim 9, comprising:determining whether a sufficient condition is met with which said set ofoperations for making two different coefficients are successfullyperformed on said basis having the minimum coefficient and said basishaving the second smallest coefficient; if not, repeating the inversionabout average operation and the selective π rotation to transform thestate into one that meets the sufficient condition; and with thesufficient condition met, sequentially performing said set ofoperations.
 11. The state configuring method according to claim 1,wherein said desired quantum mechanical state is invariant despite thesubstitution of two arbitrary sets of two-level systems and is invariantdespite the inversion between the basic and excited states of alltwo-level systems.
 12. The state configuring method according to claim11, wherein said desired quantum mechanical state is defined by afunction that outputs the number of two-level systems with said basis inthe excited state.
 13. The state configuring method according to claim11, comprising: setting a parameter for a phase rotation; providing asone unit, three continuous operations including a selective rotationoperation for selectively rotating the phase of a predetermined state insaid superposition based on said parameter, an inversion about averageoperation, and a selective rotation operation for selectively rotatingthe phase again relative to said predetermined state; and repeating saidoperation unit a predetermined times to configure said desired quantummechanical state.
 14. The state configuring method according to claim 1,wherein the Cold Trapped Ions method is used to provide ions as saidtwo-level systems, and wherein a quantized phonon mode resulting fromCoulomb interactions of the ions is used to provide external laserirradiation to allow quantum gates to perform said operations, therebyconfiguring said desired quantum mechanical state.
 15. The stateconfiguring method according to claim 14, wherein a partly entangledstate having an invariant symmetry is configured as said desired quantummechanical state, and wherein the difference in energy between the basicand first excited states of ions is precisely measured based on theRamsey spectroscopy.
 16. A communication method for encoding transmittedinformation as a quantum mechanical state configured by a stateconfiguring method according to claim 1, and executing communicationusing the quantum mechanical state as a signal on a quantumcommunication circuit.
 17. The communication method according to claim16, wherein said state configuring method is used to encode transmittedinformation as a particular partly entangled state that resists noise insaid quantum communication circuit.
 18. A state configuration apparatuscomprising: selective rotation operation means for performing aselective rotation operation on a plurality of two-level systems; andinversion about average operation means for performing an inversionabout average operation on said plurality of two-level systems, whereina desired partly-entangled quantum mechanical state, expressed by asuperposition of orthonormal bases in which each two-level systemassumes a basic or an excited state and the coefficients of the basesare all real numbers, is configured using an operation comprising acombination of an operation performed by said selective rotationoperation means and an operation performed by said inversion aboutaverage operation means.
 19. The state configuring apparatus accordingto claim 18, wherein said desired quantum mechanical state is defined bya function that causes an even number of collisions where the evennumber includes zero.
 20. The state configuring apparatus according toclaim 19, comprising the steps of: providing a first and a secondregisters for configuring a quantum mechanical state; setting aparameter for a phase rotation; writing to said second register thevalue of said predetermined function output if the value of said firstregister is input, selectively rotating the phase according to the valueof the second register and based on said parameter for a predeterminedstate in said superposition in said first register, and subsequentlyproviding as one unit, three continuous operations including a selectiverotation operation for returning the value of said second register toits initial value, an inversion about average operation that variesdepending on the value of the first register, and a selective rotationoperation for selectively rotating the phase again relative to saidpredetermined state; and repeating said operation unit a predeterminedtimes to configure the desired quantum mechanical state.
 21. The stateconfiguring apparatus according to claim 18, wherein an operationcomprising a combination of said selective rotation operation and saidinversion about average operation is performed on a flat superpositionto configure said desired quantum mechanical state.
 22. The stateconfiguring apparatus according to claim 21, wherein a Hadamardtransformation is applied to an initial state to obtain said flatsuperposition.
 23. The state configuring apparatus according to claim18, wherein said operation comprising a combination of said selectiverotation operation and said inversion about average operation is anoperation that is a combination of a series of operations forsequentially performing a selective rotation, an inversion about averageoperation, and a selective rotation; and an operation for repeating aselective n rotation and an inversion about average operation.
 24. Thestate configuring apparatus according to claim 21, wherein in saidselective rotation for transforming said flat superstition into saiddesired quantum mechanical state, the phase rotation parameter and theorder in which the bases are rotated are determined from the procedureof a transformation inverse to said transformation.
 25. The stateconfiguring apparatus according to claim 24, wherein said procedure ofinverse transformation is determined by sequentially performingoperations for making two different coefficients the same using aselective rotation operation, an inversion about average operation, anda selective rotation operation.
 26. The state configuring apparatusaccording to claim 25, comprising the steps of: rotating the phases ofbases with a negative coefficient by π to invert their sign in order tomake the coefficients of all the bases positive or zero; andsequentially performing said operations for making two differentcoefficients the same, on the basis having the minimum coefficient andthe basis having the second smallest coefficient.
 27. The stateconfiguring apparatus according to claim 26, the steps of: determiningwhether a sufficient condition is met with which said set of operationsfor making two different coefficients the same are successfullyperformed on said basis having the minimum coefficient and said basishaving the second smallest coefficient; if not, repeating the inversionabout average operation and the selective π rotation to transform thestate into one that meets the sufficient condition; and with thesufficient condition met, sequentially performing said set ofoperations.
 28. The state configuring apparatus according to claim 18,wherein said desired quantum mechanical state is invariant despite thesubstitution of two arbitrary sets of two-level systems and is invariantdespite the inversion between the basic and excited states of alltwo-level systems.
 29. The state configuring apparatus according toclaim 28, wherein said desired quantum mechanical state is defined by afunction that outputs the number of two-level systems in said basicexcited state.
 30. A state configuring apparatus according to claim 28,comprising the steps of: setting a parameter for a phase rotation;providing as one unit, three continuous operations including a selectiverotation operation for selectively rotating the phase of a predeterminedstate in said superposition based on said parameter, an inversion aboutaverage operation, and a selective rotation operation for selectivelyrotating the phase again relative to said predetermined state; andrepeating said operation unit a predetermined times to configure saiddesired quantum mechanical state.
 31. The state configuring apparatusaccording to claim 18, wherein the Cold Trapped Ions method is used toprovide ions as said two-level systems, and wherein a quantized phononmode resulting from Coulomb interactions of the ions is used to provideexternal laser irradiation to allow quantum gates to perform saidoperations, thereby configuring said desired quantum mechanical state.32. The state configuring apparatus according to claim 31, wherein apartly entangled state having an invariant symmetry is configured assaid desired quantum mechanical state, and wherein the difference inenergy between the basic and first excited states of ions is preciselymeasured based on the Ramsey spectroscopy.
 33. A communication apparatusfor encoding transmitted information as a quantum mechanical stateconfigured by a state configuring apparatus according to claim 18, andexecuting communication using the quantum mechanical state as a signalon a quantum communication circuit.
 34. The communication apparatusaccording to claim 33, wherein said state configuring apparatus is usedto encode transmitted information as a particular partly entangled statethat resists noise in said quantum communication circuit.